Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 187–238

HENCKY–PRANDTL NETS AND CONSTANT PRINCIPAL STRAIN MAPPINGS WITH ISOLATED SINGULARITIES

Julian Gevirtz

Pontifícia Universidad Católica, Facultad de Matemáticas, Casilla 306, Santiago 22, Chile Current address: 2005 North Winthrop Rd., Muncie, IN 47304, U.S.A.; jgevirtz@gw.bsu.edu

Abstract. The work presented in this paper is motivated in large measure by the appearance of Hencky–Prandtl nets (HP-nets) in the context of planar quasi-isometries with constant principal stretching factors (cps-mappings) and by compelling analogies between such mappings and those given by analytic functions of one complex variable. We study the behavior of HP-nets in the vicinity of isolated singularities and use the results of this analysis to show that if an HP-net is regular in the entire plane except for isolated singularities, then it can have at most two of them, and that all possibe nets of this kind fall into five classes each of which depends on a small number of parameters. In light of the relationship between HP-nets and cps-mappings it follows that an analogous statement holds for the latter as well, and this connection is further exploited to prove that HP-nets regular except for isolated singularities in smoothly bounded Jordan domains have nontangential limits in the appropriate sense at almost all boundary points. The treatment includes, in addition, an interpretation of cps-mappings with isolated singularities as deformations produced by the cryptocrystalline solidification with microscopic flaws of a planar film and a discussion of the problem of just how the singularities of such mappings can actually be distributed in a given domain.

Introduction

Disregarding considerations of regularity and connectivity, two mutually orthogonal one-parameter families of curves (called characteristics), covering a given plane domain D, form a Hencky–Prandtl net (abbreviated as HP-net) if for any two fixed curves C_{1} , C_{2} belonging to one of the families, the change in the inclination of the tangent is the same along all subarcs of curves of the other family which join a point of C_{1} to a point of C_{2} . Such nets are of importance in the theories of plasticity (see [Hi]) and optimal design (see [Hem]), and there is an extensive literature dealing with the analytic and numerical construction of HP-nets that satisfy various boundary conditions arising in connection with these theories as well as other applied problems. The local theory of such nets seems to have been worked out fully by Prandtl [Pra], Hencky [Hen] and Carathéodory–Schmidt [CS]. A paper of Collins [C] contains an excellent discussion of numerous aspects of the theory of Hencky–Prandtl nets and an encyclopedic bibliography. Moreover, G. Strang and

1991 Mathematics Subject Classification: Primary 73GXX, 35L99, 35L67, 30C99.

This research was supported in part by a grant from the Fundación Andes.

R.V. Kohn [SK] have described a problem which involves construction of HP-nets in both the plasticity and optimal design contexts in different parts of a given domain.

The present discussion is motivated by yet another connection in which HPnets arise, namely that of planar deformations for which the principal strains are distinct constants, henceforth called cps-mappings. Indeed, it is not difficult to show (see Proposition 1.1) that two mutually orthogonal families of curves covering a simply connected domain are the families of the lines of principal strain of a cpsmapping if and only if they form an HP-net. This class of mappings presents itself as a natural object of study in various ways.

First of all, they constitute a simple class of quasi-isometries (essentially deformations with bounded principal stretching factors) introduced and studied by F. John ([J1], [J2], [J3]). It is quite likely that for many of the as yet unresolved distortion questions for quasi-isometries raised by John, extremal behavior is displayed by cps-mappings and their higher dimensional analogues. Regardless of whether this proves to be the case, cps-mappings form a nontrivial but nonetheless tractable class of quasi-isometries whose study yields valuable insights into the extent of global distortion consistent with given bounds on local stretching.

Secondly, although governed by a nonlinear hyperbolic system (equations (1.1) in Section 1.2) cps-mappings bear, in many aspects of their behavior, notable similarities to conformal mappings, more precisely to conformal mappings f for which Re{log f′(z)} is bounded. There are several ways in which the analogy can be drawn, but for the purposes at hand it is enough to say that the function which gives the inclination of the tangent line to the curves of either of the families of the associated HP-net (to be referred to as an HP-function in the sequel) takes the role of the harmonic function arg f′(z). A simple, but striking instance of this similarity is the HP-version of Liouville’s theorem on bounded harmonic functions: an HP-function regular in the entire plane is necessarily a constant. Moreover, possibilities for developing a distortion theory for cps-mappings paralleling parts of the classical geometric theory of functions of one complex variable are described at some length in [G1, Section 4] where a numerically sharp result in this vein is established. In this paper we follow the function theory model in an investigation of isolated singularities of HP-nets and cps-mappings. In Section 1 we set down formal definitions of HP-nets and cps-mappings with minimal regularity requirements, discuss their basic properties (with proofs included for the sake of completeness), describe several procedures for the construction of HP-nets and define several specific nets which play a fundamental role in the succeeding development. The following two sections are devoted to an analysis of the behavior of HP-nets in the vicinity of an isolated singularity. Isolated singularities fall into two distinct categories depending on whether the associated HP-function is bounded on some characteristic terminating at the singular point or not; these two cases are fully analyzed in Sections 2 and 3, respectively (see Theorems 2.1 and 3.1). The results of this analysis yield information about the global consequences of the presence of a given singularity which stem from the hyperbolic nature of the underlying equations. It turns out that there are severe restrictions on the distribution of isolated singularities of an HP-net in a given domain, the exact form of which depends to a large extent on its shape—the less contorted the boundary the more difficult it is for there to be a lot of singularities. The ultimate manifestation of this phenomenon (that is, in the case in which there is no boundary) is contained in one of the main results: in Section 4 we show that an HP-net regular in the entire plane except for isolated singularities can have at most two singularities and that every such net belongs to one of five families, each of which is described by a few parameters (see Theorem 4.1). The small number of possibilities for such HP-nets is most consistent with the analogy we have been pursuing—a harmonic function whose conjugate is bounded and regular in the whole plane except for isolated singularities must be a constant. In a somewhat different direction, we use the relationship between HP-nets and cps-mappings to show in Section 5 that an HP-function regular except for isolated singularities in a smoothly bounded Jordan domain D possesses nontangential limits at almost all points of ∂D. This result closely parallels the classical Fatou theorem (see [Pri]) on the boundary behavior of bounded harmonic functions (and their conjugates). This section also contains a construction which shows that, in spite of the aforementioned limitations on the distribution of isolated singularities, on any such domain there are cps-mappings having infinitely many of them (Theorem 5.2).

Finally, deformations with constant principal strains are of concrete interest in connection with models of real situations, several of which are briefly discussed in [Y]. Consider, for example, a thin liquid film on a plane surface which upon solidification takes on a cryptocrystalline structure, that is, at each point a suitably oriented infinitesimal square of the original liquid becomes an (again, suitably oriented infinitesimal) rectangular crystal whose side lengths are constant multiples of the side length of the square. In this light global geometric results for cpsmappings acquire new significance in as much as they tell one about the extent to which the shape of the original film can change as a result of such a solidification process. Furthermore, one can interpret isolated singularities in this context as microscopic flaws in the crystallized lamina, as is explained in some detail in the final paragraph of Section 5.

1. Preliminaries

1.1. Notation and terminology. For convenience we treat the plane as C, rather than as R^{2} , and denote planar vectors as complex numbers. Let D ⊂ C be a domain and let θ be a locally Lipschitz continuous real-valued function on D. The complete integral curves of the fields e^{iθ} and ie^{iθ} will be called 1- and 2characteristics of θ, respectively. The convention that {i, j} = {1, 2} will hold throughout. Arcs of i-characteristics will be called i-arcs, or less specifically, characteristic arcs. With reference to a given θ a characteristic arc joining points a, b ∈ D will be denoted by ab and we shall use the abbreviation

Δθ(ab) = θ(b) − θ(a).

A domain Q ⊂ D will be said to be a characteristic quadrilateral of θ if ∂D is a Jordan curve lying in D containing four points a, b, c, d occurring in that order when ∂D is traversed (in either the positive or negative sense) and such that ab and cd are i-arcs and bc and da are j -arcs. We will refer to such a Q as abcd and use the abbreviation

Δ^{2}(abcd) = Δθ(bc) − Δθ(ad) = Δθ(dc) − Δθ(ab).

Furthermore, D_{1} and D_{2} will denote differentiation with respect to arc length in the directions e^{iθ} and ie^{iθ}, respectively; that is, for differentiable u

D_{1}u(z) = cos(θ(z))u_{x}(z) + sin(θ(z))u_{y}(z),
D_{2}u(z) = − sin(θ(z))u_{x}(z) + cos(θ(z))u_{y}(z).

We use the symbol λ(E) to denote the 1-dimensional measure of the set E, so that in particular λ(C) is the length of the simple arc C.

1.2. Definition and basic properties of HP-nets and cps-mappings. In dealing with HP-nets it is frequently more convenient to work with the function θ which gives the inclination of the tangent to the curves belonging to one or the other of the two families that make up the net, rather than with the net itself. Since, however, we shall be working with domains that are not simply connected, an inevitable contingency in any discussion of singularities, a minor complication arises; namely, that upon going around a hole θ may change its value by a multiple of π, (not 2π). Thus, in the following definition we consider functions which, although not necessarily single-valued in the entire domain D under consideration, do have a single-valued branch on any simply-connected subdomain of D.

Definition 1.1. Let D ⊂ C be a domain. A (possibly multivalued) function θ on D is called an HP-function if it satisfies the following conditions:

(i) Every point p in D has a neighborhood E on which θ has a Lipschitz continuous branch and satisfies Δ^{2}(abcd) = 0 for all characteristic quadrilaterals abcd of θ contained in E.

(ii) e^{2iθ} is single-valued in D. The set of all HP-functions on D will be denoted by HP(D).

It is necessary to consider e^{2iθ} in (ii), rather than e^{iθ}, since, as noted above, in going around a hole in D, θ might change by a multiple of π. We will use the term HP-net to refer to the families of integral curves of the fields e^{iθ} and ie^{iθ}; through each point of D there passes exactly one curve of each family. It is evident that the corresponding net remains unchanged if we add \frac{1}{2}π to an HP-function; this causes, of course, an interchange of the two families of characteristics. We use the notation HP(D) to denote the set of all HP-nets, as well as the set of all HP-functions, on D; this minor ambiguity will cause no confusion.

To discuss cps-mappings we need the following notation:

T (θ) = [ \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ]  and  S(m_{1}, m_{2}) = [ \begin{matrix}m_{1} & 0 \\ 0 & m_{2} \end{matrix} ] ,

where here and in what follows m_{1} and m_{2} are distinct positive numbers.

Definition 1.2. A mapping of a domain D ⊂ C into C is called an (m_{1}, m_{2})- mapping if each point of D has a neighborhood N in which there are two Lipschitz continuous functions θ = θ_{f} and ϕ = ϕ_{f} such that the Jacobian matrix J_{f} of f is given by T (−ϕ)S(m_{1}, m_{2})T (θ).

If θ and ϕ are Lipschitz continuous in a simply-connected domain D, then T (−ϕ)S(m_{1}, m_{2})T (θ) is the Jacobian matrix of a mapping if and only if

(1.1)  D_{1}(m_{1}θ − m_{2}ϕ) = 0 and D_{2}(m_{1}ϕ − m_{2}θ) = 0 a.e. in D,

or in other words

(1.2)  m_{i}θ − m_{j}ϕ is constant along i-arcs of θ.

To see this, it is enough to show, in light of the Lipschitz continuity of θ and ϕ, that these conditions simply amount to the formal necessary and sufficient compatibility conditions on the entries of a matrix in order for it to be the Jacobian of a mapping. For a given fixed p ∈ D, let θ_{0} = θ(p) and ϕ_{0} = ϕ(p), and let D_{1}u, D_{2}u denote the directional derivatives of u at p in the directions e^{iθ_{0}} and ie^{iθ_{0}} , respectively. Then functions A and B give D_{1}u and D_{2}u if and only if D_{2}A = D_{1}B. (Note that at p D_{1}D_{2}u is not the same as D_{1}D_{2}u since the latter involves derivatives of θ and the former does not. The compatibility conditions can, of course, be formulated in terms of D_{1}D_{2}u and D_{2}D_{1}u—see (1.4) below— and we shall make subsequent use of that formulation also.) Let f = u+iv. Then J_{f} = T (−ϕ)S(m_{1}, m_{2})T (θ) is equivalent to

[ \begin{matrix}D_{1}u & D_{2}u \\ D_{1}v & D_{2}v \end{matrix} ] = T (−ϕ)S(m_{1}, m_{2})T (θ)T (−θ_{0}) = T (−ϕ)S(m_{1}, m_{2})T (θ − θ_{0}).

The compatibility conditions for this matrix are equivalent to those for any left multiple by a constant invertible matrix, a convenient choice in this case being T (ϕ_{0}). If we write θ̄ = θ − θ_{0} and ϕ̄ = ϕ − ϕ_{0} then the compatibility conditions simply state that the result of applying D_{1} to the second column of T (−ϕ̄)S(m_{1}, m_{2})T (θ̄) is the same as that of applying D_{2} to the first column. Since θ̄ and ϕ̄ are both 0 at p and

T (−ϕ̄)S(m_{1}, m_{2})T (θ̄)
= [ \begin{matrix}m_{1} cos ϕ̄ cos θ̄ + m_{2} sin ϕ̄ sin θ̄ & m_{1} cos ϕ̄ sin θ̄ − m_{2} sin ϕ̄ cos θ̄ \\ m_{1} sin ϕ̄ cos θ̄ − m_{2} cos ϕ̄ sin θ̄ & m_{1} sin ϕ̄ sin θ̄ + m_{2} cos ϕ̄ cos θ̄ \end{matrix}] ,

a trivial calculation shows that at p, and therefore at any point, the compatibility equations take the form (1.1) as desired.

Equations (1.1) constitute the nonlinear hyperbolic system alluded to at the beginning of the fourth paragraph of the introduction. We next have

Proposition 1.1. Let D be a simply connected domain and m_{1} , m_{2} distinct positive numbers. Then θ = θ_{f} for some (m_{1}, m_{2})-mapping f of D if and only if θ is an HP-function on D.

Proof. Let θ and ϕ be the functions associated with an (m_{1}, m_{2})-mapping f of D and let abcd ⊂ D be a characteristic quadrilateral with 1-sides ab, dc. Then from (1.2) we have

Δ^{2}ϕ(abcd) = Δϕ(dc) − Δϕ(ab) = \frac{m_{1}}{m_{2}} (Δθ(dc) − Δθ(ab)) = \frac{m_{1}}{m_{2}} Δ^{2}θ(abcd).

But if we write Δ^{2}ϕ(abcd) as Δϕ(bc)−Δϕ(ad), we see that Δ^{2}ϕ(abcd) also equals (m_{2}/m_{1})Δ^{2}θ(abcd), so that indeed Δ^{2}θ(abcd) = 0.

Conversely, given that θ is an HP-function, let Q be a closed characteristic quadrilateral in D and let p be an interior point of Q. Then, since Δ^{2}θ(Q′) = 0 for all characteristic quadrilaterals Q′ ⊂ Q, it is clear that once ϕ(p) = ϕ_{0} has been assigned, there is a unique ϕ in Q which satisfies (1.2). This ϕ can then be extended bit by bit “from one characteristic quadrilateral to the next” to all of D so that (1.2) holds, that the resulting ϕ is single-valued follows from the simple connectedness of D via the monodromy principle. By what was established above the matrix T (−ϕ)S(m_{1}, m_{2})T (θ) is the Jacobian of an (m_{1}, m_{2})-mapping of D, as desired.

It is clear that given an (m_{1}, m_{2})-mapping on a simply connected domain D, the (continuous) HP-function θ = θ_{f} is uniquely determined to within an additive constant (which is a multiple of π), and that all (m_{1}, m_{2})-mappings g of D for which θ_{g} = θ_{f} are of the form e^{iα}f + z_{0} , α ∈ R, z_{0} ∈ C. The i-characteristics are the curves along which f changes arc length by a factor of m_{i}.

For a function θ ∈ C^{2}(D) straightforward formal calculation shows that each of the equations

(1.3)  D_{j}D_{i}θ = (−1)^{j}(D_{i}θ)^{2}

when written in terms of differentiation in the x and y directions takes the form

\frac{1}{2}(sin 2θ)(θ_{yy} − θ_{xx}) + (cos 2θ)θ_{xy} = (cos 2θ)(θ_{x}^{2} − θ_{y}^{2}) − 2(sin 2θ)θ_{x}θ_{y},

so that each case of (1.3) implies the other. The meaning of equations (1.3) can be expressed in the following geometric form. If κ_{i}(z) denotes the unsigned curvature of the i-characteristic C_{i}(z) through z, then the derivative of κ_{i} in the direction orthogonal to C_{i}(z) towards its concave side is κ_{i}^{2} . Of key importance in what is to follow is the fact that the solution of the equation κ′(t) = κ^{2}(t), with κ(0) = κ_{0} is κ(t) = κ_{0}/(1−κ_{0}t). Equations (1.3) consequently imply that if κ_{i} ≠ 0 it increases as we move along C_{j}(z) towards the concave side of C_{i}(z) and decreases as we move along C_{j}(z) in the opposite direction. In particular the length of any j -arc emanating from z towards the concave side of C_{i}(z) is at most 1/κ_{i}(z).

Proposition 1.2. Let D be a simply connected domain and θ ∈ C^{2}(D). Then θ is an HP-function if and only if equations (1.3) hold on D.

Proof. Let θ ∈ C^{2}(D). Straightforward calculations show that functions A, B ∈ C^{1}(D) are of the form A = D_{1}u and B = D_{2}u for some u ∈ C^{2}(D) if and only if

(1.4)  D_{2}A − D_{1}B = AD_{1}θ + BD_{2}θ

holds. That is, the compatibility conditions may be expressed in this form (see the discussion following (1.2)). Assume that θ is an HP-function and let m_{1} , m_{2} be distinct positive numbers. Let ϕ be a function (whose existence was established in the preceding proposition) for which (1.1) holds. Applying (1.4) first with A = D_{1}θ, B = D_{2}θ and then with

A = D_{1}ϕ = \frac{m_{1}}{m_{2}} D_{1}θ, B = D_{2}ϕ = \frac{m_{2}}{m_{1}} D_{2}θ,

we have

(1.5)  D_{2}D_{1}θ − D_{1}D_{2}θ = (D_{1}θ)^{2} + (D_{2}θ)^{2},

and

(1.6)  \frac{m_{1}}{m_{2}} D_{2}D_{1}θ − \frac{m_{2}}{m_{1}} D_{1}D_{2}θ = \frac{m_{1}}{m_{2}} (D_{1}θ)^{2} + \frac{m_{2}}{m_{1}} (D_{2}θ)^{2},

from which the equations (1.3) follow immediately.

Conversely, let θ ∈ C^{2}(D) satisfy (1.3). We define

P = \frac{m_{1}}{m_{2}} D_{1}θ  and  Q = \frac{m_{2}}{m_{1}} D_{2}θ.

Then it follows from (1.3) that

D_{2}P − D_{1}Q = P D_{1}θ + QD_{2}θ,

so that there is a function ϕ satisfying D_{1}ϕ = P and D_{2}ϕ = Q; that is, the function ϕ satisfies (1.1). Thus θ is an HP-function by the preceding proposition.

In order to proceed with the development of the local analytic aspects of the theory of HP-nets and cps-mappings, and to have an important tool for the construction of such we need to introduce characteristic coordinate mappings. Let θ be an HP-function on a domain D. Let I_{i} = [a_{i}, b_{i}], τ_{i} ∈ I_{i}, i = 1, 2. Let p ∈ D and let z = z_{i}(s), s ∈ I_{i} be an arc length parametrization of the i-arc through p with z_{i}(τ_{i}) = p, i = 1, 2. For (t_{1}, t_{2}) ∈ S = I_{1} × I_{2} let ζ(t_{1}, t_{2}) be the point common to the 1-characteristic through z_{2}(t_{2}) and the 2-characteristic through z_{1}(t_{1}). For λ(I_{1}) and λ(I_{2}) sufficiently small ζ: S → D is a bi-Lipschitz homeomorphism, as follows from the Lipschitz continuity of θ and simple facts about the dependence of solutions of ordinary differential equations on initial values. Without loss of generality we can assume that |θ(z)| < \frac{1}{4}π in the corresponding characteristic quadrilateral ζ(S), so that in what follows all arguments lie in the interval (−\frac{1}{4}π, \frac{3}{4}π). We write α_{i}(s) = arg\{z′_{i}(s)\}. That θ is an HP-function is equivalent to

(1.7)  ω(t_{1}, t_{2}) = θ(ζ(t_{1}, t_{2})) = α_{1}(t_{1}) + α_{2}(t_{2}) − α_{2}(τ_{2}).

Because θ is Lipschitz continuous, α_{i} is differentiable a.e. on I_{i} and α′_{i} is a bounded measurable function. If ζ = ξ + iη, then the functions ξ, η satisfy the system

(1.8)  ξ_{t_{1}} sin ω − η_{t_{1}} cos ω = 0;  ξ_{t_{2}} cos ω + η_{t_{2}} sin ω = 0.

Writing

v = −ξ sin ω + η cos ω  and  u = ξ cos ω + η sin ω,

that is, u + iv = ζe^{−iω}, the system (1.8) takes the form

(1.9)  u_{t_{2}} = α′_{2}(t_{2})v;  v_{t_{1}} = −α′_{1}(t_{1})u.

This is a very simple hyperbolic system which becomes even more transparent when expressed in integral form

\begin{aligned}v(t_{1}, t_{2}) & = v_{0}(t_{2}) − ∫_{τ_{1}}^{t_{1}} u(τ, t_{2})α′_{1}(τ) dτ, \\ u(t_{1}, t_{2}) & = u_{0}(t_{1}) + ∫_{τ_{2}}^{t_{2}} v(t_{1}, τ)α′_{2}(τ) dτ, \end{aligned}

where

(1.10)  \begin{aligned}u_{0}(t_{1}) & = Re\{z_{1}(t_{1})e^{−iα_{1}(t_{1})}\}, \\ v_{0}(t_{2}) & = Im\{z_{2}(t_{2})e^{−i(α_{2}(t_{2})+α_{1}(τ_{1})−α_{2}(τ_{2}))}\}. \end{aligned}

Clearly, u_{0} and v_{0} are Lipschitz continuous.

A straightforward and standard argument based on iteration shows that given continuous u_{0} and v_{0} this system of integral equations has a global solution u, v ∈ C(S) which consequently satisfies (1.9) almost everywhere; the solution is, moreover, unique. Furthermore, if the initial data as well as the functions α_{1} , α_{2} are C^{∞}, then the solution is likewise C^{∞} on S; this standard regularity result stems from the fact that the derivatives of u and v satisfy a system of the same general form. Consequently, if we start with C^{∞} arc length parametrizations z = z_{i}(s), s ∈ I_{i} with

(1.11)  z_{1}(τ_{1}) = z_{2}(τ_{2})  and  z′_{2}(τ_{2}) = iz′_{1}(τ_{1}),

then we will obtain a C^{∞} mapping ζ: S → C. It is easy to see that if this mapping is one-to-one, then the images of the lines t_{i} =const form an HP-net on ζ(S); that is, that θ(z) = ω(ζ^{−1}(z)) will be an HP-function. In general, of course, ζ will not even be locally one-to-one, because the images of lines t_{i} =const can cross. However, if we are given an upper bound K for the curvatures of the initial curves z_{i}(I_{i}), then equations (1.3) allow one to deduce that there exist δ = δ(K) and L = L(K) such that ζ will be one-to-one in N = [τ_{1} −δ, τ_{1} −δ]×[τ_{2} −δ, τ_{2} −δ]∩S, and that the corresponding θ will satisfy a Lipschitz condition with constant L in ζ(N). From this, via a simple approximation procedure and a compactness argument, one can show that the same is true if one only assumes that the functions arg z_{i} satisfy Lipschitz conditions with constant K (instead of the initial curves being C^{∞} with curvatures bounded by this constant). Summarizing, we have the following

Proposition 1.3. If z = z_{i}(s), s ∈ I_{i}, i = 1, 2 are arc length parametrizations for which the arg{z′_{i}(s)} are Lipschitz continuous and satisfy (1.11), then there is some neighborhood N of (τ_{1}, τ_{2}) in S on which ζ is one-to-one and such that θ(z) = ω(ζ^{−1}(z)) is an HP-function on ζ(N).

If ζ is one-to-one on all of S, as will be the case when the C_{i} = z_{i}(I_{i}) are adjacent sides of a characteristic quadrilateral of an already existing HP-function, then we shall refer to the curves ζ(I_{1} × {t}), t ∈ I_{2} as translates of C_{1} along each ζ({t} × I_{2}), t ∈ I_{1} and call them parallel arcs, and analogously when the roles of the indices 1, 2 are reversed. We shall also refer to C_{i} and C_{j} as being perpendicular or orthogonal to each other. We shall refer to the uniquely defined net given by θ(z) = ω(ζ^{−1}(z)) in the image of any neighborhood of (τ_{1}, τ_{2}) in which ζ is one-to-one as HP(C_{1}, C_{2}).

Proposition 1.4. If, in addition to the hypotheses of Proposition 1.3, we assume that arg{z′_{1}(s)} is nonincreasing on I1^{+} = {t ∈ I_{1} : t ≥ τ_{1}} and arg{z′_{2}(s)} is nondecreasing on I2^{+} = {t ∈ I_{2} : t ≥ τ_{2}}, then ζ is locally one-to-one on I1^{+} ×I2^{+} and θ(z) = ω(ζ^{−1}(z)) is an HP-function on ζ(J) for any open J ⊂ I1^{+} × I2^{+} on which ζ is one-to-one.

Proof. The hypotheses imply that z_{1}(I1^{+}) is convex toward its left-hand side (i.e., toward the side corresponding to increasing t_{2} ) and that z_{2}(I2^{+}) is convex toward its right-hand side (i.e., toward the side corresponding to increasing t_{1} ). In the C^{∞} case this means, in light of the significance of equations (1.3) (see the paragraph immediately preceding Proposition 1.2), that both families of characteristics diverge, that is, that the curvatures of the i-characteristics decrease with increasing t_{j}. From this the locally one-to-one character of ζ follows immediately. (The mapping might fail to be globally one-to-one due to the possibility that ζ might not give a simple covering of ζ(I1^{+} × I2^{+}).) The general case follows by approximating the z_{i} by sequences {z_{i,n}(s)} of C^{∞} arc length parametrizations for which the arg{z′_{i,n}(s)} have the stipulated monotonicity as well as uniformly bounded derivatives which tend to d arg{z′_{i}(s)}/ds in measure.

Next, we explain the sense in which equations (1.3) hold for general (i.e., not necessarily C^{2} ) HP-functions θ. We define E_{i} = E_{i}(θ) to be the set of all points p such that if z = z(s), −ε < s < ε, with z(0) = p is an arc length parametrization of an i-arc of θ containing p, then θ(z(s)) is differentiable at s = 0. Obviously, almost all points (with respect to arc length) of each i-characteristic belong to E_{i} and almost all points of the domain on which θ is defined (with respect to 2-dimensional measure) belong to E_{1} ∩ E_{2} .

Proposition 1.5. Let θ be an HP-function on D and let C_{k}, k = 1, 2, be the k -characteristic through p ∈ E_{i}. Then C_{j} ⊂ E_{i}, and equation (1.3) holds along C_{j}, when D_{j} is iterpreted as arc length differentiation along C_{k} in the direction i^{k−1}e^{iθ}, k = 1, 2.

Proof. Without loss of generality we assume, for definiteness, that p ∈ E_{1} . Let z = z_{i}(s), −α ≤ s ≤ α, i = 1, 2 be arc length parametrizations of small pieces of C_{1} and C_{2} with z_{i}(0) = p and with the directions of increasing s correspond to e^{iθ} and ie^{iθ}, respectively. Let κ_{i}(s) = d arg{z′_{i}(s)}/ds. Let ζ be the corresponding characteristic coordinate mapping and let F_{i}(t_{1}, t_{2}) denote the translate of z_{i}([0, t_{i}]) along C_{j} from p to z_{j}(t_{j}). Since κ_{0}/(1 − tκ_{0}) is the solution of the initial value problem κ′ = κ^{2} , κ(0) = κ_{0} , it is clearly enough to show that for each t ∈ (0, α]

(1.12)  lim_{s→0^{+}} \frac{θ(ζ(s, t)) − θ(ζ(0, t))}{λ(F_{1}(s, t))} = \frac{κ_{1}(0)}{1 − tκ_{1}(0)}.

That (1.12) also holds for negative t and as s → 0^{−}, can be deduced with minor notational adjustments to the argument to follow. Because of trivial compactness considerations the size of α > 0 is not important, so that we may assume that

(1.13)  sup\{|κ_{i}(s)| : i = 1, 2, |s| ≤ α\} ≤ \frac{1}{100α}.

Here “sup” is to be interpreted as “essential supremum”. This condition implies that the F_{i}(t_{1}, t_{2}) are very close to being straight line segments, and in addition that if we start with C^{∞}z_{i} satisfying it, then the corresponding characteristic coordinate mapping will be one-to-one in [−α, α] × [−α, α]. We fix such an α. Very simple estimates show that

lim_{s→0} λ(F_{2}(s, t)) = t, 0 < t ≤ α,

uniformly over the class of all ζ arising from C^{∞}z_{i} satisfying (1.13).

The equation D_{2}D_{1}θ = (D_{1})^{2} , expressed in terms of the radius of curvature R_{1} = 1/κ_{1} , says that D_{2}R_{1} = −1. Now, for such C^{∞} initial curves, a simple calculus argument involving an appropriate Riemann sum and passage to the limiting integral, together with this differential equation for R_{1} shows that

λ(F_{1}(s, t)) = ∫_{0}^{s} 1 − κ(σ)λ(F_{2}(σ, t)) dσ,

so that

\begin{aligned}λ(F_{1}(s, t)) & = ∫_{0}^{s} 1 − κ(σ)(t + o(1)) dσ = ∫_{0}^{s} 1 − κ(σ)t dσ + o(s) \\ & = s − t(θ(z_{1}(s)) − θ(z_{1}(0))) + o(s), \end{aligned}

where the “little-o” is uniform over the entire C^{∞} class indicated above. From the HP-property we have that

θ(ζ(s, t)) − θ(ζ(0, t)) = θ(z_{1}(s)) − θ(z_{1}(0)).

Abbreviating this difference by Δ we see that the difference quotient on the lefthand side of (1.12) is equal to

\frac{Δ}{s − tΔ + o(s)} = \frac{Δ/s}{1 − tΔ/s + o(1)}.

By approximation by C^{∞} functions as in the proof of the preceding proposition we have that the same holds for the original HP-net. But Δ/s → κ_{1}(0), so that (1.12) is indeed true.

Because of Proposition 1.5 the comments contained in the paragraph immediately preceding Proposition 1.2 are relevant in the context of general (i.e., not necessarily C^{2} ) HP-nets and their content will play a key role in much of what follows. For convenience we formulate the next proposition, which gives a lower bound for the area of a characteristic quadrilateral, in terms of the characteristic coordinate mapping ζ discussed above.

Proposition 1.6. Let the mapping ζ be one-to-one on all of I_{1} × I_{2} , where I_{i} = [τ_{i}, σ_{i}]. If the lengths of all of the translates of C_{2} = z_{2}(I_{2}) along C_{1} = z_{1}(I_{1}) are at least m, then the area of ζ(I_{1} × I_{2}) is at least \frac{1}{2}mλ(C_{1}).

Proof. Again, by a straightforward approximation procedure we can reduce consideration to the case in which the arc length parametrizations z_{1} , z_{2} are C^{∞}. Consider a small subarc J = z_{1}([σ, σ + δ]) of C_{1} . The length of J is obviously δ and the length of its translate t units along ζ({σ} × I_{2}) is easily found (see the proof of the preceding proposition) to be (1 − tκ_{1}(σ))δ + O(δ^{2}), where κ_{1}(σ) = d arg z_{1}(σ)/dσ. For small δ the translates of J are virtually straight line segments orthogonal to the curve ζ({σ} × I_{2}). Since the lengths of the 2-arcs of the characteristic quadrilateral in question are all at least m, equations (1.3) imply that κ_{1}(σ) ≤ 1/m, so that the area of ζ{[σ, σ + δ] × I_{2}} is then seen to be at least

δ ∫_{0}^{m}(1 − \frac{t}{m}) dt + O(δ^{2}) = \frac{δm}{2} + O(δ^{2}),

so that upon considering the appropriate Riemann sum and passing to the limiting integral, the area of ζ(I_{1} × I_{2}) is indeed at least \frac{1}{2}mλ(C_{1}).

We define Di^{+}θ(p) to be the upper limit of |D_{i}θ(z)| as z → p. We have the following simple consequence of Proposition 1.5.

Proposition 1.7. Let θ be an HP-function on D. (i) If the j -characteristic C through p is a simple curve, then

Di^{+}θ(p) ≤ \frac{1}{L} ≤ \frac{1}{dist (p, ∂D)},

where L is the length of the shorter of the two arcs into which p divides C. (ii) If the j -characteristic through p is a closed curve then Di^{+}θ(p) = 0.

Proof. We prove (i), the proof of (ii) involving only minor variations. For definiteness and without loss of generality we assume that i = 1. We can also assume that Di^{+}θ(p) > 0, since otherwise there is nothing to prove. From the definition of Di^{+}θ(p) it follows that for any ε > 0 there is a point q ∈ E_{1}(θ) within ε of p for which |D_{1}θ(q)| > Di^{+}θ(p) − ε and such that there are 2-arcs C_{+} and C_{−} emanating from q in the directions ie^{iθ(q)} and −ie^{iθ(q)} , respectively, which have length greater than L − ε. Assume that D_{1}θ(q) is positive. Then applying Proposition 1.5 at the point z(s) which lies s units from q along C_{+} we have that

D_{1}θ(z(s)) = \frac{D_{1}θ(q)}{1 − sD_{1}θ(q)}.

Thus

L − ε < λ(C_{+}) ≤ \frac{1}{D_{1}θ(q)} < \frac{1}{Di^{+}θ(p) − ε};

that is, that Di^{+}θ(p) − ε < 1/(L − ε), which establishes the desired bound. If D_{1}θ(q) < 0 then one arrives at the same conclusion by following C_{−} instead of C_{+} .

As an immediate corollary of Proposition 1.7 we have

Proposition 1.8. If θ is an HP-function on all of C, then Di^{+}θ(p) = 0 for all p ∈ C, i = 1, 2, so that θ is a constant.

Proposition 1.9 (Compactness principle). For any domain D the family {e^{2iθ} : θ ∈ HP(D)} is compact in the topology of uniform convergence on compact subsets of D.

Proof. This follows via elementary arguments, since Proposition 1.7 implies that if U ⊂ D is a closed disk, then θ satisfies a Lipschitz condition with constant at most 1/ dist(U, ∂D).

We end this subsection with the discussion of an important limiting case of the characteristic coordinate mapping construction we have been using, namely that in which one of the initial curves degenerates to a point. Let z = z(s), s ∈ I_{1} = [0, τ_{1}] be an arc length parametrization with Lipschitz continuous derivative, and let I_{2} be an interval one of whose endpoints is 0 and whose length is less than 2π. Let ω(t_{1}, t_{2}) = arg{z′(t_{1})} + t_{2} . We consider the same system (1.8) of differential equations as before, but with initial conditions corresponding to ζ(t_{1}, 0) = z(t_{1}) and ζ(0, t_{2}) = z(0), that is, for the functions u, v defined by u + iv = ζe^{−iω}, the equations are

u_{t_{2}} = v  and  v_{t_{1}} = −α′(t_{1})u,

where α(t) = arg{z_{1}(t)} with the corresponding initial conditions

u_{0}(t_{1}) = Re{z(t_{1})e^{−iα(t_{1})}}  and  v_{0}(t_{2}) = Im{z(0)e^{−iα(0)−t_{2}}}.

This characteristic initial value problem, in light of the discussion preceding Proposition 1.4, is well-posed.

Proposition 1.10. The mapping ζ defined immediately above exists on I_{1} ×I_{2} and is one-to-one on J = [0, ε]×I_{2} for some ε > 0. Moreover, the function θ(z) = ω(ζ^{−1}(z)) is an HP-function on the interior of ζ(J).

Proof. This may be proved in a fashion directly analogous to that in which Proposition 1.2 was justified, or alternatively by applying the compactness principle to the family of HP-functions resulting from the original (i.e., nondegenerate) characteristic coordinate mapping construction with z_{1}(s) = z(s), s ∈ I_{1} and

z_{2}(s) = z(0) + δ(e^{is/δ} − 1)z′(0), s ∈ δI_{2} = {δs : s ∈ I_{2}} = I_{2}(δ).

The curve given by z_{2} is an arc of a circle of radius δ orthogonal to z(I_{1}) at z(0) for which z′(0) is an outward pointing normal. One then lets δ → 0 and obtains the desired result by the compactness principle together with the convexity of the curve z_{2}(I_{2}(δ)).

It is to be noted that the characteristic arcs ζ({t_{1}} × I_{2}), t_{1} ∈ (0, ε] are convex (with their concave side towards z(0)). The family of orthogonal arcs ζ((0, ε] × {t_{2}}), t_{2} ∈ I_{2} is a fan of characteristic arcs which are confluent at z(0). If ζ is one-to-one in all of the rectangle, and the curve parametrized by z is C, then we will denote the resulting uniquely defined net by Fan(C, I_{2}).

1.3. Isolated singularities of HP-functions and cps-mappings. Henceforth the r-neighborhood of a point p ∈ C will be denoted by N(p, r), and N(p, r)\{p} will be denoted by N′(p, r). If p is a point of the domain D, an HP-function θ on D\{p} is said to have an isolated singularity at p. The point p will be called a true singularity of θ if θ cannot be extended to an HP-function in D. We use the terms “singularity” and “true singularity” for HP-nets also. An HP-function (HP-net) on D\A for some set A of isolated points of D will be called an HP*-function (HP*-net) on D; HP*(D) will denote both the class of HP*-functions and that of HP*-nets on D. Furthermore, we shall denote by cps*(D) (by cps*(D, m_{1}, m_{2})) the set of all cps-mappings ((m_{1}, m_{2})-mappings) which are defined on a set of the form D\A, where A is a set of isolated points of D, and whose continuous extensions to D are local homeomorphisms.

Proposition 1.11. If θ has a true singularity at p, then the essential supremum of ∇θ is not finite in N′(p, ε), for any ε > 0.

Proof. If the essential supremum of ∇θ is finite in some such punctured neighborhood N′(p, ε), then θ is single valued and Lipschitz continuous there, and consequently can be extended by continuity to all of N(p, ε) with the same Lipschitz constant. If abcd is any characteristic quadrilateral whose closure lies in the punctured neighborhood, then Δ^{2}θ(abcd) = 0. By a trivial limit argument it then follows that this is true even if p is on the boundary of the quadrilateral. If abcd is a characteristic quadrilateral of θ which contains p in its interior, then there are points a′ ∈ ab, b′ ∈ bc, c′ ∈ cd, and d′ ∈ da such that a′c′ and b′d′ are characteristic arcs of θ passing through p. But then

Δ^{2}θ(abcd) = Δ^{2}θ(a′bb′p) + Δ^{2}θ(pb′cc′) + Δ^{2}θ(d′pc′d) + Δ^{2}θ(aa′pd′) = 0,

so that θ is an HP-function in N(p, ε), contrary to the hypothesis.

Proposition 1.12. Let θ be an HP-function which has a true singularity. Then there is a characteristic arc of θ of finite length one of whose endpoints is p.

Proof. Let θ be defined in N′(p, ε). From the preceding proposition it follows that there are points q ≠ p in N(p, ε/2) such that Di^{+}θ(q) > 2/ε for at least one of i = 1 or 2. But then it follows from Proposition 1.7 that there is a j -arc of length at most ε/2 which joins q to p.

Proposition 1.12 suggests the following classification of true singularities.

Definition 1.2. A true singularity of an HP-function θ is said to be a singularity of type R or a Riemann singularity if θ is bounded on some characteristic arc of finite length which terminates at p. Otherwise it is said to be of type S or a spiral singularity.

The reason we have chosen the names “Riemann” and “spiral” for the two kinds of singularities will be made clear in what follows (see (i) and (ii) in the following subsection, and Theorems 2.1 and 3.1). Here again, we apply these terms to HP-nets also.

1.4. HP-nets with one or two singularities. In this paragraph we define four families of HP-nets; one of the main goals of this paper is to prove that apart from the trivial nets corresponding to constant θ, these are the only nets in HP*(C).

(i) Spiral nets. For p ∈ C, −\frac{1}{2}π ≤ α ≤ \frac{1}{2}π, we define

σ_{p,α}(p + re^{iϕ}) = ϕ + α.

To see that θ = σ_{ρ,α} is an HP-function, let r = |z − p|. Simple trigonometry shows that D_{1}θ(z) = sin α/r and D_{2}r = − sin α, from which it follows that D_{2}D_{1}θ(z) = (sin α/r)^{2} = (D_{1}θ(z))^{2} , so that σ_{ρ,α} is indeed an HP-function by Proposition 1.2. It is equally easy to see that for 0< |α| < \frac{1}{2}π we have that σ_{p,α}(z) → ±∞ as z → p along any i-characteristic (−∞, if 0< α and i = 1 or α < 0 and i = 2; +∞ otherwise), so that the characteristics spiral around p. Quite specifically, for p = 0 the polar equations of the 1- and 2-characteristics through the point r_{0}e^{iθ_{0}} are

(1.14)  r = r_{0}e^{(ϕ−θ_{0}) cot α}  and  r = r_{0}e^{−(ϕ−θ_{0}) tan α},

respectively. The values α = 0, ±\frac{1}{2}π give rise to the degenerate case in which the families of 1-characteristics consist of rays emanating from p (when α = 0) and circles centered at p (when α = ±\frac{1}{2}π). The net corresponding to σ_{p,α} will be denoted by 𝒮_{p,α}.

Let f ∈cps*(C, m_{1}, m_{2}) for which θ_{f} = σ_{p,α}. Then the curves f(C) are congruent for all i-characteristics C, from which it follows that f(N(p, δ)) is a disk N(f(p), δ′). Simple trigonometry implies that f changes arc length on circles

∂N(p, δ) by a factor of \sqrt{m_{1}^{2} sin^{2} α + m_{2}^{2} cos^{2} α}. Since f changes area by a factor of m_{1}m_{2} we have

m_{1}m_{2}πδ^{2} = π(δ\sqrt{m_{1}^{2} sin^{2} α + m_{2}^{2} cos^{2} α} )^{2},

so that if m_{1}/m_{2} = μ, we have

μ sin^{2} α + \frac{1}{μ} cos^{2} α = 1,

from which it follows that μ = cot^{2} α. Since μ ≠ 0, 1, we see that in order for f to exist, |α| must be in (0, \frac{1}{2}π)\\{\frac{1}{4}π\}. The argument just given is easily seen to be reversible, that is, for all such α there is an (essentially unique) (m_{1}, m_{2})-mapping for which θ_{f} = σ_{p,α}, provided that m_{1}/m_{2} = cot^{2} α.

If for 0 < |α| < \frac{1}{2}π we follow the two characteristics of σ_{p,α} through z_{0} = p + re^{iθ_{0}} as they move toward p, they cross infinitely often. For convenience let 0 < α < \frac{1}{2}π. Then a very simple calculation based on (1.14) shows that they meet for the first time at the point a = p + z_{0}e^{T (i−tan α)} , where T = 2π cos^{2} α. Let C_{i} denote the i-arc joining z_{0} to a. Arcs C_{1} and C_{2} together form a simple closed curve encircling p and are both convex toward the outside of this curve. The interior angles at z_{0} and a are seen to be \frac{1}{2}π and \frac{3}{2}π, respectively, so that if β_{i} is the (unsigned) change in θ along C_{i} we have

(1.15)  β_{1} + β_{2} = 2π.

(ii) Riemann nets. For p ∈ C, α real and 0 ≤ β ≤ π, we define

ρ_{p,α,β}(p + re^{iϕ}) = \{ \begin{matrix}α, & α ≤ ϕ ≤ α + \frac{1}{2}π, \\ ϕ − \frac{1}{2}π, & α + \frac{1}{2}π ≤ ϕ ≤ α + β + \frac{1}{2}π, \\ α + β, & α + β + \frac{1}{2}π ≤ ϕ ≤ α + β + π, \\ ϕ − π, & α + β + π ≤ ϕ < α + 2π, \end{matrix}

and then define ρ_{p,α,β}(p + re^{i(ϕ+2πn)}) to be ρ_{p,α,β}(p + re^{iϕ}) − nπ, for n = ±1, ±2, …. It is easy to verify that the multivalued function ρ_{p,α,β} is indeed an HP-function in C\{p}; in each of the four sectors it coincides with one of the degenerate cases of σ_{p,α} or is constant, and it is continuous (modulo π). Quite specifically, in the four sectors on the right-hand side of the definition of ρ_{p,α,β} the 1-characteristics are, respectively, the rays {p + sie^{iα} + te^{iα} : t ≥ 0}, s ≥ 0, circular arcs with center at p, rays {p + sie^{i(α+β)} − te^{i(α+β)} : t ≥ 0}, s ≥ 0, and rays {p+te^{iϕ} : t ≥ 0}, α+β +π ≤ ϕ ≤ α+2π. We also note that ρ_{p,α,β} increases by π along any simple closed curve which goes around p in the positive direction and that ∇ρ_{p,α,β} has jumps along the rays that separate the four sectors (because along these rays characteristics of one or the other of the families change from straight lines to circular arcs). The net corresponding to ρ_{p,α,β} will be denoted by ℛ_{p,α,β}. We have chosen the term “Riemann nets”, because their restrictions to half-planes arise in connection with certain “Riemann problems” for the hyperbolic system (1.1). A trivial calculation based on (1.1) shows that ρ_{p,α,β} is θ_{f} for some f ∈cps*(C, m_{1}, m_{2}) if and only if (m_{1}/m_{2})β + (m_{2}/m_{1})(π − β) = π (since otherwise f(N(p, δ)) would not give a simple covering of a neighborhood of f(p)).

The remaining two nets are special cases of the following general construction; no simple formula for the corresponding HP-functions would appear to be available. Let I_{k} = [0, t_{k}] and z_{k}: I_{k} → C be arc length parametrizations of the curves C_{k} with Lipschitz continuous derivatives for which z_{1}(0) = z_{2}(0) and z_{1}(t_{1}) = z_{2}(t_{2}). Assume furthermore that arg{z′_{1}(s)} and arg{z′_{2}(s)} are nonincreasing and nondecreasing, respectively, that z_{1}(0), z_{1}(t_{1}) are the only two points that these curves have in common, and that

z′_{2}(0) = iz′_{1}(0)  and  z′_{2}(t_{2}) = iz′_{1}(t_{1}).

One sees that the simple closed curve C_{1} ∪C_{2} is the boundary of a “heart-shaped” domain whose “point” is at z_{1}(t_{1}) = z_{2}(t_{2}). Now apply the characteristic coordinate construction of Section 1.2. By Proposition 1.4 and simple geometry it follows that the characteristic coordinate mapping is one-to-one on [0, t_{1}) × [0, t_{2}). Let E(C_{1,}C_{2}) = ζ([0, t_{1})×[0, t_{2})), Z(C_{1}) = ζ([0, t_{1}]×{t_{2}}), Z(C_{2}) = ζ({t_{1}}×[0, t_{1}]). From the convexity of the original curves C_{1} and C_{2} together with the assumption that they meet at right angles at both endpoints, it easily follows that E(C_{1,}C_{2}) lies in the complement of the interior of the simple closed curve C_{1} ∪ C_{2} and that Z(C_{1}) and Z(C_{2}) satisfy exactly the same conditions as the original curves C_{1} and C_{2} did. We inductively define C_{i}^{(0)} = C_{i}, C_{i}^{(k+1)} = Z(C_{i}^{(k)}), i = 1, 2, k = 0, 1, 2, …. It is then easy to see that the HP-nets so defined in the interiors of the E(C_{1,}^{(k)}C_{2}^{(k)}), k = 0, 1, 2, …, fit together to form a single HP-net in the interior of their union, that is, in the doubly connected domain which constitutes the exterior of the original simple closed curve C_{1} ∪ C_{2} .

Figure 1.

(iii) Double Riemann nets. We shall define nets 𝒟_{p,q,α,β,γ} where p ≠ q are points, 0 ≤ β, γ ≤ π, α real, and 0 < arg(e^{−iα}(q − p)) < \frac{1}{2}π. They are called double Riemann nets because in neighborhoods of p and q they coincide with ℛ_{p,α,β} and ℛ_{q,α+π,γ}, respectively. The construction is facilitated by reference to Figure 1, in which α = −\frac{1}{2}π, and 0 < β, γ < π. Changing α simply requires a rotation of the figure; the cases in which β or γ is 0 or π require self-evident modifications which are left to the reader. This picture, in which some characteristic arcs have been drawn, is merely descriptive and is not meant to show exactly what these arcs look like, but only their general form.

Since the nets ℛ_{p,α,β} and ℛ_{p,α+π,γ} coincide in the rectangle prqs, the two together give an HP-net in the curvilinear polygon whose sides (listed in positive order) are the 2-arc ap′′, the 2-arc p′′r, the 1-arc rq′, the 1-arc q′b, the 2arc bq′′, the 2-arc q′′s, the 1-arc sp′, and the 1-arc p′a (except, of course, at the singularities p and q). Note that the only jumps in the argument of the tangent when this simple closed curve is traversed in the positive direction are jumps of \frac{1}{2}π at a and b and −\frac{1}{2}π at r and s. Using the characteristic coordinate mapping construction and Proposition 1.4, it is clear that this net can be extended to the heart shaped region bounded by the union of the curves C_{1} and C_{2} which are made up of the arcs rp′′, p′′a, ac, and rq′, q′b, bc, respectively. The construction described in the paragraph immediately preceding this discussion of double Riemann nets can then be used to extend this net to the entire complement of {p, q}. It is obvious that the only singularities of this net are the ones of type R at p and q.

(iv) Degenerate double Riemann nets. We shall define nets ℱ_{p,q,σ,β,γ} where p ≠ q are points, 0 ≤ β, γ ≤ π, and σ = + or −. These nets arise as limiting cases of double Riemann nets as arg(e^{−iα}(q − p)) tends to 0 or \frac{1}{2}π. The description is again facilitated by reference to the corresponding Figure 2 in which σ = +, and 0 < β, γ < π. The value of σ indicates the sign of arg((p′ −p)/(q −p)) ∈ [−π, π]. The cases in which β or γ is 0 or π require self-evident modifications which are left to the reader, and similarly for the case σ = −. The net is defined initially as ℛ_{p,arg(q−p)−π/2,β} and ℛ_{q,arg(p−q)−π/2,γ} inside the circular sectors pqp′ and qpq′, respectively. One then uses the fan construction summed up in Proposition 1.10 to define Fan(pq′, [β − π, 0]) and Fan(qp′, [γ − π, 0]), which gives an HP-net in the interior of the curvilinear polygon pwq′qup′p. One then extends this net by tacking on HP(qq′w, qu) (that is, the characteristic quadrilateral qwvu), so that the net is now defined in the interior of the heart-shaped region bounded by the mutually orthogonal characteristics pp′ ∪ p′u ∪ uv and pw ∪ wv (except at the singularities p and q). Finally, the construction given just before the discussion of double Riemann nets is applied to define the net in the entire complement of {p, q}. The case σ = − is the same except that (p′ − p)/(q − p) and (q′ − q)/(p − q) lie in the lower half-plane.

2. Riemann singularities

We shall establish a series of lemmas which lead to the complete description of singularities of type R given in Theorem 2.1. The reader is reminded that λ denotes 1-dimensional measure and that the term “translate” is used in the sense explained in the paragraph immediately following the statement of Proposition 1.3. We begin with

Figure 2.

Lemma 2.1. Let θ have a singularity of type R at p. Then there is a characteristic arc of θ, one of whose endpoints is p and along which θ(z) has a limit as z tends to p.

Proof. From the definition of this type of singularity it follows that we can assume that θ is regular in N = \overline{N(p, δ)}\\{p\}, and that there is an i-arc C of θ of finite length which joins a point lying outside of \overline{N(p, δ)} to p and on which θ is bounded. Let w = w(s), 0 ≤ s ≤ δ, with w(δ) = p be the arc length parametrization of a final segment of C. Let θ(s) = θ(w(s)). The curvature κ(s) = θ′(s), exists for s ∈ A, where λ(A) = δ. We may assume that lim_{s→δ} θ(s) does not exist, since otherwise C is itself a characteristic arc of the kind we are seeking. For any κ_{0} > 0 and 0 ≤ δ′ < δ we have

λ({s ∈ (δ′, δ) ∩ A : κ(s) > κ_{0}}) > 0

and

λ({s ∈ (δ′, δ) ∩ A : κ(s) < −κ_{0}}) > 0,

since if either of these sets had measure zero for any such κ_{0} and δ′, the boundedness of θ(s) on (0, δ) would imply the existence of lim_{s→δ} θ(s). From this it is easy to see that there is an s_{1} ∈ (δ/2, δ) such that κ(s_{1}) > 2/δ, and s_{1} is a density point of {s ∈ A : κ(s) > 0}. The part C_{1} of the j -characteristic emanating from w(s_{1}) to the left of C (as it is traversed in the direction of increasing s) has length at most δ/2 (by Proposition 1.5), and so lies entirely in the punctured neighborhood N and terminates at p. Similarly, there is an s_{2} ∈ (s_{1}, δ) for which κ(s_{2}) < −2/δ, and which is a density point of {s ∈ A : κ(s) < 0}. The part C_{2} of the j -characteristic emanating from w(s_{2}) to the right of C again has length at most δ/2, and so lies entirely in the punctured neighborhood N and terminates at p. Also, the opposite signs of the curvature of C at the points w(s_{1}) and w(s_{2}), together with simple topological considerations show that C_{1} ∩ C_{2} = {p}. Now, we do not know a priori that C_{1} does not intersect C at more than one point, so we let s′_{1} < s_{2} be the greatest number less than s_{2} such that w(s′_{1}) ∈ C_{1} and we let C′_{1} denote that part of C_{1} which connects w(s′_{1}) to p. (The arc C′_{1} might, of course, be all of C_{1} .) It follows from the fact that p is the only point common to C_{1} and C_{2} and the fact that the lengths of these curves as well as that of C are all less than δ, that C′_{1} ∪ w([s′_{1}, s_{2}]) ∪ C_{2} is a simple closed curve whose interior D lies entirely in the punctured neighborhood N of p; the exterior of D is easily seen to lie to the right as w([s′_{1}, s_{2}]) is traversed from s′_{1} to s_{2} . We denote by E the part of the j -characteristic emanating to the left of C from w(s_{2}); obviously, E ⊂ D. Let v = v(s), 0 ≤ s < L be the arc length parametrization of E, with v(0) = w(s_{2}). (We are not a priori excluding the possibility that L = ∞.) Since we chose s_{2} to be a density point of {s ∈ A : κ(s) < 0}, there is a positive ε < δ − s_{2} , s_{2} − s′_{1} such that

(2.1)  λ(\{s ∈ I ∩ A : κ(s) < 0\}) > \frac{1}{2}ε,

both when I = I_{1} and I = I_{2} , where I_{1} and I_{2} are the left and right halves of the interval J = [s_{2} − ε, s_{2} + ε]. Let L′ > 0 denote the supremum of all s ∈ (0, L) for which the translates of w(J) along E down to v(s) exist. (Here again we are not assuming a priori that L′ is finite, although we shall show shortly that this must be the case.) These translates of w(J) are all contained in D. For t ∈ [−ε, ε] and s ∈ [0, L′) we let v(t, s) be the point which lies both on the translate of w(J) through v(s) and on the translate of an initial arc of E through w(s_{2} + t). Now, it follows from (2.1) that for s ∈ [0, L′) the lengths of the translates of w(I_{1}) and w(I_{2}) down to v(s), are both at least ε/2, which together with Proposition 1.6 implies that \frac{1}{2}εL′ ≤ πδ^{2} . Thus, L′ is finite. Also, Proposition 1.7 implies that Dj^{+}(v (s)) ≤ 2/ε, for s ∈ (0, L′). This, together with the finiteness of L′ means that lim_{s→L′} θ(v(s)) exists, so that by the defining property of HP-nets, lim_{s→L′} θ(v(t, s)) exists for each t ∈ J. This then says that lim_{s→L′} v(t, s) exists for each t ∈ J. But the definition of L′ implies that for some t = t_{0} ∈ J this limit is p. The j -arc parametrized by v(t_{0}, s), 0 ≤ s < L′, can therefore be taken as the desired arc.

Lemma 2.2. Let θ be regular in N = N′(p, δ). Let C_{k} be a k -arc of θ lying in N and with arc length parametrization z_{k}(s), 0 ≤ s < L_{k}, k = 1, 2. Let lim_{s→L_{i}} z_{i}(s) = p and lim_{s→L_{i}} θ(z(s)) exist. Let z_{i}(0) = z_{j}(0). Then either (i) for some ε > 0 all translates of C_{i} along z_{j}([0, ε]) lie in N and terminate at p, or (ii) for some ε > 0 the translate of z_{j}((0, ε]) along C_{i} provides a j -arc terminating at p, along which θ(z) has a limit as z tends to p and which is orthogonal to C_{i} at p.

Proof. By replacing C_{i} by a sufficiently small final subarc, and δ by a smaller number if necessary, we may assume that the values of θ on C_{i} lie in some small interval, say of length 1/100. Let η > 0 be so small that the values of θ(z_{j}(s)) on [0, η] also lie in an interval of length 1/100 and all translates of z_{j}([0, η]) along C_{i} lie in N. Let E_{s} denote the translate of C_{i} with initial point z_{j}(s), and let p(s) denote its terminal point. If for some s_{0} ∈ (0, η], p(s_{0}) = p, then the small variability of θ on z_{j}([0, η]) implies that the length of the translates of z_{j}([0, s_{0}]) down to z_{i}(s) tend to 0 as s tends to L_{i}. This means that (i) holds with ε = s_{0} . In the opposite case p(s), 0 < s ≤ η, parametrizes a j -arc for which lim_{s→0} θ(p(s)) exists. Now, because of the small variability of θ on C_{i}, sup{λ(E_{s}) : 0 ≤ s < L_{i}} = L < 3δ. It is easy to show by C^{∞} approximation and simple calculus that the length of all translates of z_{j}([0, s]) along C_{i} is at most s + Lξ(s), where ξ(s) is the total variation of θ on z_{j}([0, s]). Since s + Lξ(s) → 0 as s → 0, it follows that p(s) → p as s → 0, so that (ii) is true with ε = η. The orthogonality of p([0, η]) to C_{i} at p follows immediately from the fact that ξ(s) → 0.

Lemma 2.3. Let θ be regular in N = N′(p, δ). Let z(s), 0 ≤ s < L be an arc length parametrization of an i-arc C of θ lying in N. Let lim_{s→L} z(s) = p and let lim_{s→L} θ(z(s)) exist. Then one of the following must happen:

(i) There is an open j -arc containing some point of C such that none of the translates of C along this arc terminates at p.

(ii) Passing through some point of C there is a closed j -arc C′ of positive length such all translates of C along C′ terminate at p, but on one side of each end point of C′ no nearby translates of C terminate at p. Furthermore, the former translates are mutually nontangential at p.

(iii) θ(z) is given by arg(z − p) or arg(z − p) ± \frac{1}{2}π (to within an integral multiple of π) in a punctured neighborhood of p.

Proof. By replacing δ by a smaller number if necessary, we may assume that θ is regular in N′(p, 6δ), that C joins ∂N(p, 6δ) to p and that the values of θ on C all lie in a very small interval, of length 1/100, say. Let q denote the point of C for which |q − p| = δ. To facilitate the exposition we assume that j = 1 and that at q the unit tangent to C pointing towards p is ie^{iθ(q)} . Let w(s), K < s < K′, be the arc length parametrization of the j -characteristic in N′(p, 3δ) for which w(0) = q and such w′(s) = e^{iθ(w(s))} . (It is possible that this j -characteristic is a simple closed curve or that it has infinite length; to cover these possibilities we allow either one or both of K, K′ to be infinite.) Assume that (i) does not hold. Let ℐ denote the set of all subintervals I of (K, K′) which contain 0 and are such that for all s in I, w(s) is joined to p by an i-arc C_{s} in N′(p, 2δ) (which is just a translate of the part of C which joins q to p). By Lemma 2.2, ℐ contains intervals of positive length. Let I ∈ ℐ . By the defining property of HP-nets, the set {θ(z) : z ∈ C_{s}} is the same for all s ∈ I, and by assumption it is an interval of length less than 1/100. From this it follows that

(2.2)  |arg \frac{w′(s)}{i(w(s) − p)}| < 1/100,  s ∈ I,

and that

(2.3)  |w(s) − p| ≤ λ(C_{s}) ≤ 2|w(s) − p|,  s ∈ I.

(The constant 2 is, of course, much larger than necessary.) It furthermore follows from the fact that all of the C_{s}, s ∈ I, meet at p, together with Proposition 1.5 that,

(2.4)  dθ(w(s))/ds = 1/λ(C_{s}),  a.e. on I.

Indeed, were this not the case, a simple argument shows that there would be a positive distance between two of the arcs C_{s}, which is not consistent with their all meeting at p. Let ϕ(s) denote the argument of the tangent (the one pointing away from p, to be definite) to C_{s} at p. Note that by the defining property of HP-nets all of the C_{s} have tangents at p because the original C did. A straightforward argument based on this property shows that ϕ′(s) = dθ(w(s))/ds. From this, together with (2.3) and (2.4) it follows that

(2.5)  ϕ′(s) = dθ(w(s))/ds ≥ 1/(2|w(s) − p|)  a.e. on I.

If λ(θ(w(I))) ≤ 2π, it follows from (2.2) and the fact that |q − p| = δ that

(2.6)  δ/2 < |w(s) − p| < 2δ,  s ∈ I,

and in particular that \overline{w(I)} ⊂ N′(p, 3δ).

First, assume that sup{λ(θ(w(I))) : I ∈ ℐ } < 2π. Then it is easy to see that ∪{I : I ∈ ℐ } = I_{0} ∈ ℐ is closed. This gives the first sentence in (ii); the second sentence follows from the lower bound for ϕ′(s) in (2.5).

To complete the proof it therefore suffices to assume that sup{λ(ϕ(I)) : I ∈ ℐ } ≥ 2π. If this is the case, then there is an I = [α, β] ∈ ℐ , such that for all s in I, w(s) is joined to p by an i-characteristic arc C_{s} in N′(p, 3δ) and such that ϕ(β) = ϕ(α) + 2π. But then it follows that either C_{α} and C_{β} coincide or one is a proper subarc of the other, since distinct characteristics cannot be tangent at p, as follows immediately from the positiveness of ϕ′(s). First we consider the case that they coincide, that is, that w([α, β]) is a simple closed convex curve (recall that arg(w′(s)) is nondecreasing) containing p in its interior. Let w(γ) be the point on this curve at maximum distance from p. Then Dj^{+}θ(w(γ)) ≥ 1/|w(γ) − p|, so that by Proposition 1.7, C_{γ} must be a straight line segment, from which it follows that all of C_{s} are straight line segments. This means that there are straight line j -characteristics terminating at p at all angles ϕ, 0 ≤ ϕ ≤ 2π, so that (iii) holds. The other possibility cannot, in fact occur. To see this, we assume for definiteness that C_{β} is a proper subarc of C_{α}. Thus w([α, β]) starts at a point w(α) on C_{α}, goes around p once and ends up at a point w(β) on C_{α} between w(α) and p. But then it is easy to see that on [α, K′), w(s) winds around p infinitely many times in the counterclockwise direction, crossing C_{α} successively at points w(s_{n}), n = 0, 1, 2, …, each of which lies farther along C_{α} towards p than its predecessor. Since the arcs w(s_{n})w(s_{n+1}) are translates of each other, if there were any variation of θ on the j -arc w(s_{0})w(s_{1}), θ(z) would not have a limit as z → p along C_{α}, an obvious contradiction, since C_{α} is parallel to a terminal segment of our original i-arc C. But then the arcs w(s_{n})w(s_{n+1}) are all line segments of the same length, which is obviously impossible.

Definition 2.1. In the case that (ii) of the preceding lemma holds, the family F (C) of translates of the i-arc C joining the points of the j -arc C′ to p will be called an i-fan of the net at p, and the characteristics passing through the endpoints of C′ are called the bounding characteristics of F (C). The angle formed at p by the bounding characteristics is called the angle of the fan. In the case that (i) holds we regard C alone as constituting a (degenerate) fan with angle 0 and both bounding characteristics coinciding with C.

In dealing with such fans we will usually restrict attention to a small neighborhood of p so that on C, θ takes values in a small interval (taken to have length less than 1/100, to be specific). In this way all of the i-arcs making up an i-fan are virtually straight line segments and join p to the boundary of the disk about p in which we are working. These i-arcs are connected by almost circular j -arcs whose curvature increases uniformly to infinity as they approach p.

With these preliminaries out of the way we are in a position to derive a description of singularities of type R. For the sake of descriptive simplicity we assume, without loss of generality, that p = 0. We apply Lemma 2.1 and then Lemma 2.3 and assume that we are not in case (iii) of the latter. There is therefore an i-fan of characteristic arcs terminating at 0 (which might consist of a single arc—case (i) of Lemma 2.3), which, without loss of generality, we assume to be “symmetric” with respect to the positive x-axis, that is, that the bounding characteristics Ci^{+} and Ci^{−} have outgoing tangents which make angles of ±\frac{1}{2}ϕ, respectively, with the positive x-axis. As we saw in the proof of Lemma 2.3, 0 ≤ ϕ < 2π (since we are not in case (iii)). By choosing an appropriate δ > 0, we can assume that θ is regular in N′(0, 2δ), and that along each of the characteristics of the fan the values of θ lie in an interval of length 1/100, say. From the definition of fan in conjunction with Lemma 2.2, we see that there are distinct j -characteristics Cj^{+} and Cj^{−} emanating from 0, whose outgoing tangents at 0 form angles of ±\frac{1}{2}π with those of Ci^{+} and Ci^{−}, respectively. By picking a smaller δ, if necessary, we can assume that along each of these j -characteristics the values of θ lie in an interval of length 1/100. We note the following.

(1) ϕ ≤ π. This is clear since were π < ϕ < 2π, nearby translates of Cj^{+} would cross nearby translates of Cj^{−} in N′(0, 2δ) and form an angle strictly between 0 and π, a patent impossibility.

(2) If ϕ = 0, the fans F (Cj^{+}) and F (Cj^{−}) cannot both consist of a single characteristic. To see that this is so, assume otherwise. Then there would be an i-characteristic, E, tangent to the x-axis and emanating to the left of 0. We can assume as before that along E the values of θ lie in an interval of length 1/100. But this means that through each point of Ci^{+} = Ci^{−}, Cj^{+} , Cj^{−} and E there passes a perpendicular characteristic which joins two points of ∂N(0, 2δ) without passing through p. This, in turn, implies that |∇θ(z)| ≤ 2/δ, a.e. in N′(0, δ), which, by Proposition 1.11, is impossible, because we are assuming that the net has a true singularity at 0.

(3) F (Cj^{+}) = F (Cj^{−}). This is obvious if ϕ = π, since then Cj^{+} and Cj^{−} are tangent at 0 and consequently coincide (see (ii) of Lemma 2.3), and the corresponding fan is degenerate. To handle the case in which 0 ≤ ϕ < π, we assume to the contrary that these fans are disjoint. Let E^{+} and E^{−} be the other bounding j -characteristics of F (Cj^{+}) and F (Cj^{−}), respectively. Then it follows from (2), or the condition 0 < ϕ < π, whichever is applicable, that the angle between these distinct j -characteristics is strictly between 0 and π. But then by the definition of fan there are i-characteristics perpendicular to E^{+} and E^{−}, respectively, which intersect in N′(0, δ) and form an angle strictly between 0 and π, which is obviously absurd.

We are now in a position to state in the form of several theorems the basic facts about Riemann singularities which we shall use in what follows.

Theorem 2.1. Let p be a Riemann singularity of an HP-function θ. Then there exists a δ > 0 such that one of the following holds.

(A) θ(z) is given by arg(z − p) or arg(z − p) ± \frac{1}{2}π (to within an integral multiple of π) in N′(p, δ).

(B) There are four arcs Ci^{+} , Cj^{+} , Cj^{−}, Ci^{−} with the following properties:

(1) Each one joins ∂N(p, δ) to p.

(2) The arc length parametrization each of them has a Lipschitz continuous derivative.

(3) Ci^{+} and Cj^{+} have their single common point at p, where they meet at right angles, and similarly for Ci^{−} and Cj^{−}.

(4) Ck^{+} and Ck^{−} either have their single common point at p or coincide, k = 1, 2.

(5) The four arcs occur in the indicated order in the counterclockwise sense.

(6) If α_{k} ∈ [0, π] denotes the (unsigned) angle between Ck^{−} and Ck^{+} , k = 1, 2, then in the smaller curvilinear sector of N(p, \frac{1}{2}δ) between Ci^{−} and Ci^{+} the characteristics of the net coincide with those of Fan(Ci^{−}, [0, α_{i}]) and in the smaller curvilinear sector of N(p, \frac{1}{2}δ) between Cj^{+} and Cj^{−} they coincide with those of

Fan(Cj^{+}, [0, α_{j}]).

(7) In the two parts into which N(0, \frac{1}{2}δ) is divided by the removal of these fans, the characteristic arcs of the net coincide with those of HP(Ci^{+}, Cj^{+}) and HP(Cj^{−}, Ci^{−}), respectively.

(8) α_{i} + α_{j} = π.

Proof. If (A) (which is (iii) of Lemma 2.3) does not occur, then it follows from the preceding discussion that there is one fan of i-arcs and one of j -arcs, and that at most one of these fans can be degenerate. If, as above, we assume that θ is regular in N′(0, 2δ), and that along each of the curves belonging to these fans the values of θ lie in a small interval (say of length 1/100), then, by Proposition 1.7, the curvature on the parts of these curves lying in N′(0, δ) is bounded above by 1/δ, since for any point on any these curves there is an orthogonal characteristic arc of length at least δ emanating to either side and lying in N′(0, 2δ); this proves point (2). Points (3) and (4) follow from the preceding discussion and point (5) is merely of a notational nature. Point (6) follows from Proposition 1.10 and the discussion preceding it. It follows from point (2) and the discussion preceding Proposition 1.4 that in each of the two parts into which N(0, δ) is divided by the removal of the fans, θ is the solution of the characteristic initial value problem corresponding to bounding characteristics of the two fans; that is, in those regions the HP-net is given by HP(Ci^{+}, Cj^{+}) and HP(Cj^{−}, Ci^{−}), respectively. That the angles of the fans sum to π, follows from the preceding discussion.

Definition 2.2. In case (A) of the preceding theorem the singularity p will be called a degenerate spiral singularity; in case (B) it will be called a nondegenerate Riemann singularity.

The following theorem is essentially the converse of Theorem 2.1; its proof is an immediate consequence of the constructions of HP-nets given in Section 1.2.

Theorem 2.2. Let C_{1} , C_{2} be closed arcs with bounded curvature (i.e., whose arc length parametrizations have Lipschitz continuous first derivatives) with an endpoint p in common and which form a right angle at p. Let ϕ_{1}, ϕ_{2} ≥ 0 and ϕ_{1} +ϕ_{2} = π. Then in some punctured neighborhood N of p there is a unique HPnet which coincides with HP(C_{1}, C_{2}) in the part of N in the smaller curvilinear sector determined by these curves and which has fans F (C_{1}) and F (C_{2}) with angles ϕ_{1} , ϕ_{2} , respectively.

As an immediate consequence of the foregoing discussion we also have the following

Theorem 2.3. Let p be a Riemann singularity of an HP-net. Let C_{i} be an i-characteristic arc which terminates at p and let C_{j} be an open j -arc which intersects C_{i}. If all of the translates of C_{i} along C_{j} exist and terminate at p, then C_{j} is strictly concave towards p.

Here we mean, of course, that if z(s), a < s < b, is an arc length parametrization of C_{j} for which iz′(s) points in the direction along i-arcs towards p, then the essential infimum of d arg z′(s)/ds is strictly positive. The following theorem follows immediately from Theorem 2.1 together with Proposition 1.5.

Theorem 2.4. Let p be a Riemann singularity of an HP-net for which the angle of the i-fan is π and let E_{1} and E_{2} be the bounding characteristic arcs of the i-fan at p. Then E_{1} and E_{2} cannot be both concave or both convex towards the (unique) j -characteristic terminating at p.

We end this section with

Theorem 2.5. If ℋ is an HP-net regular in C except for a single nondegenerate Riemann singularity at p, then ℋ is an ℛ_{p,α,β}.

Proof. It is clearly sufficient to prove that the fans at p consist entirely of straight lines. If this were not so, then one, and hence all of the characteristics in the i-fan, say, would have nonvanishing curvature on some set of positive measure. Thus, there would be a j -arc C_{j} which joins a point q on a bounding characteristic C_{i} of the i-fan to p. The i-arc E joining q to p (including the endpoints) together with C_{j} forms a simple closed curve. Simple topological considerations show that either C_{j} is an initial arc of one of the bounding characteristics of the j -fan at p, or one of these bounding characteristics has an initial arc lying in the interior of the simple closed curve E ∪ C_{j} and joining p to some point q′ on E. But then consideration of nearby translates of E produces an i-arc joining two points of a j -characteristic, yielding an i-arc and a j -arc joining a pair of distinct points and thereby forming a simple closed curve on and inside of which the net is regular, a patent absurdity.

3. Spiral singularities

Now that we have completely described Riemann (type R) singularities, we deal with the remaining possibility, spiral or type S singularities. In this section we shall consequently assume that p is an isolated singularity of an HP-function such that if C is a characteristic arc joining q ≠ p to p, then either C has infinite length or θ(z) is unbounded as z tends to p along C.

Lemma 3.1. Let θ be regular in N′(p, 2δ) and have a spiral singularity at p. Let C be an i-arc of θ lying in N′(p, δ), one of whose endpoints is p. Then C is not tangent to any of the concentric circles ∂N(p, d), 0 < d < δ.

Proof. Without loss of generality we may assume that p = 0. Assume, to the contrary, that z(s), 0 ≤ s < L, is the arc length parametrization of a subarc of C′ of C with lim_{s→L} z(s) = 0, such that z′(0) · z(0) = 0, where for simplicity we regard complex numbers as vectors and use the dot notation for inner product. Let w(s) = z′(s) · z(s). Since z(s) tends to 0 as s tends to L, lim_{s→L} w(s) = 0. Thus there is a point σ ∈ (0, L) at which |w| attains its maximum. Arbitrarily close to σ there are numbers s_{1} < s_{2} such that

0 = w(s_{2}) − w(s_{1}) = (z′(s_{2}) − z′(s_{1})) · z(s_{2}) + z′(s_{1}) · (z(s_{2}) − z(s_{1})).

Dividing by s_{2} −s_{1} , and taking into account that z′ is continuous and |z′(σ)| = 1, we see that |z(σ)|Di^{+}(z(σ)) ≥ 1. But then it follows from Proposition 1.7 that z(σ) is joined to p by a j -arc which is a straight line segment. But this contradicts the hypothesis that p is a spiral singularity.

The following lemma is similar to Lemma 2.2; we have chosen not to combine the two in a single more general lemma, since to do so requires some discussion not necessary for the purposes at hand.

Lemma 3.2. Let θ be regular in N′(p, 2δ) and have a spiral singularity at p. Let C_{k} be a k -arc of θ lying in N′(p, δ) with arc length parametrization z_{k}(s), 0 ≤ s < L_{k}, k = 1, 2. Let lim_{s→L_{i}} z_{i}(s) = p. Let z_{i}(0) = z_{j}(0). Then for some ε > 0 each point of z_{j}([0, ε]) is joined to p by an i-arc lying in N(p, δ).

Proof. We begin with the following observation. Let w_{n}(s), 0 ≤ s ≤ α_{n}, n = 1, 2, …, be arc length parametrizations of j -arcs E_{n} ⊂ N′(p, δ) of θ for which λ(E_{n}) ≥ ξ > 0 and λ(θ(E_{n})) < 1/100. Because of the latter condition the E_{n} are close to being line segments, so that in particular ξ ≤ λ(E_{n}) ≤ 3δ for all n ≥ 1. Thus, the sequence of functions v_{n}(t) = w_{n}(λ(E_{n})t), 0 ≤ t ≤ 1, is uniformly bounded and equicontinuous, and consequently has a subsequence which converges uniformly on [0, 1] to some function v. A straightforward argument shows that either v(t), 0 ≤ t ≤ 1, gives a parametrization of a j -arc of θ in \overline{N(p, δ)}\\{p\} with one endpoint possibly coinciding with p, or there is a t_{0} ∈ (0, 1) such that v, when restricted to each of the intervals [0, t_{0}] and [t_{0}, 1], parametrizes such a j -arc. Furthermore, the values of θ on any of these arcs parametrized by v lie in an interval of length at most 1/100.

By replacing C_{j} with an initial subarc, if necessary, we may assume that λ(θ(C_{j})) < 1/100 and that C_{j} intersects C_{i} only at z_{i}(0) = z_{j}(0). For s ∈ (0, L_{i}) and η ∈ (0, L_{j}] let C(s, η) denote the translate of the subarc z_{j}([0, η]) of C_{j} along C_{i} down to z_{i}(s), provided it exists and lies in N′(p, δ). For each η ∈ (0, L_{j}] let L(η) be the supremum of all s ∈ [0, L_{i}) such that C(s, η) exists. Obviously, L(η) > 0 for all η ∈ (0, L_{j}]. We claim that there is some η > 0 for which L(η) = L_{i}. This could only fail to be true if for each η ∈ (0, L_{j}] there is an increasing sequence {s_{m}(η)} in [0, L(η)) which tends to some σ(η) < L_{i} such that one of the following happens:

(i) dist(C(s_{m}(η), η), ∂N(p, δ)) → 0 or

(ii) dist(C(s_{m}(η), η), p) → 0.

First we show that (i) cannot happen for arbitrarily small η. Say, to the contrary, that (i) actually happens for η = η_{n}, where η_{n} → 0. Then, since λ(θ(z_{j}([0, η]))) → 0 as η → 0, the observation of the first paragraph of the proof will produce a straight line j -arc E joining ∂N(p, δ) to some point of C_{1} ∪ {p}. But this contradicts the hypothesis that p is a spiral singularity, since there would then be a straight line j -arc (a translate of an initial segment of E) of length at least dist(C_{1}, ∂N(p, δ)) > 0 emanating from each point of C_{1} ∪ {p}. Thus there is some η_{0} > 0 such that (i) cannot happen if η ≤ η_{0} . Now let η ≤ η_{0} and assume that (ii) happens. But then, again by the initial observation, there will be a j -arc J joining z_{i}(σ(η)) to p for which λ(θ(J)) < 1/100, which again contradicts the hypothesis that p is a spiral singularity. Thus, indeed, there is some ε > 0 such that C(s, ε) is well defined for all s ∈ [0, L_{i}). This, of course, is the same as saying that all the translates of C_{i} along z_{j}([0, ε]) exist and lie in N′(p, δ). To finish we only have to show that λ(C(s, ε)) → 0 as s → L_{i}. But if this were not true we would have a sequence {s_{m}} in [0, L_{i}) tending to L_{i} for which λ(C(s_{m}, ε)) ≥ ξ > 0, and the observation with which we began would give us a j -arc J of length at least ξ terminating at p = lim_{m→∞} z_{i}(s_{m}) for which λ(θ(J)) ≤ 1/100, which again contradicts the assumption that θ has a spiral singularity at p.

Lemma 3.3. Let θ be a HP-function with a type S singularity at p. Then there exists some δ_{1} > 0 such that θ is regular in N′(p, 2δ_{1}) and such that each point q in N′(p, δ_{1}) is joined to p in N(p, |q −p|) by characteristic arcs from both families neither of which is tangent to ∂N(p, |q − p|).

Proof. Let θ be regular in N(p, δ), and let C be an i-arc lying in N(p, δ), with arc length parametrization z(s), 0 ≤ s < L, and which terminates at p. We begin by observing that (the measurable function) |dθ(z(s))/ds| must be unbounded on [0, L). Assume, to the contrary, that it is bounded. Then L must be ∞, since if L < ∞, lim_{s→L} θ(s) would exist, which would contradict the hypothesis that p is a spiral singularity. But if |dθ(z(s))/ds| is bounded and L = ∞, then clearly lim_{s→L} z(s) cannot exist. Thus indeed |dθ(z(s))/ds| is unbounded on [0, L). Proposition 1.7 then implies that there exists a δ_{1} in (0, \frac{1}{2}δ) such that some point q of ∂N(p, δ_{1}) ∩ C is joined by a j -arc lying in N(p, \frac{1}{2}δ). A simple application of Lemma 3.1 shows that we can assume in addition that this j -arc as well as a subarc C′ of C join q to p inside \overline{N(p, δ_{1})}\\{p\}. For 0 < ε ≤ δ_{1} let A_{i}(ε) be the set of z ∈ ∂N(p, ε) which can be joined to p by an i-arc lying in \overline{N(p, ε)}\\{p\}. From what we have shown it follows that A_{i}(ε) ≠ ∅, i = 1, 2, 0 < ε ≤ δ_{1} .

We show that A_{i}(ε) is open in ∂N(p, ε). If there is an i-arc which joins a point z of the circle ∂N(p, ε) to p in \overline{N(p, ε)}, then it follows immediately from Lemma 3.2 that there is a neighborhood S of z on this circle such that all w in S can also be joined to p by an i-arc lying in N(p, \frac{1}{2}δ). Moreover, it follows from Lemma 3.1 that these i-arcs are not tangent to the circle and that they lie in \overline{N(p, ε)}.

Next we show that ∂N(p, ε)\A_{i}(ε) is also open in ∂N(p, ε). If z ∈ ∂N(p, ε) is not joined to p by an i-arc lying in \overline{N(p, ε)}, then we claim that if we proceed along the i-characteristic through z in either direction, we will be led outside of \overline{N(p, ε)}. Assume, to the contrary, that we can proceed indefinitely along an i-arc C ⊂ \overline{N(p, ε)}, beginning at z but without coming to p. Then C has infinite length, and if w(s), 0 ≤ s < ∞, is the arc length parametrization of C, then lim_{s→∞} w(s) is not p. In consequence, if ξ is the infimum of all η such that N(p, η) contains w((s, ∞)) for some s, then 0 < ξ ≤ ε. Let δ > ξ′ > ξ > ξ′′ > 0. Then for some s′, w((s′, ∞)) lies in N(p, ξ′), but for arbitrarily large s, w(s) lies outside of N(p, ξ′′). From this it easily follows that there are arbitrarily large s for which w(s) lies in N(p, ξ′)\N(p, ξ′′), but such that Di^{+}θ(w(s)) > 1/ξ′. However, for such s there is a j -arc of length less than ξ′ which joins w(s) to p, by Proposition 1.7. Obviously, the length of such an arc must be at least ξ′′, so that these arcs come closer and closer to straight line segments as ξ′ and ξ′′ approach ξ. By a simple limit argument (see the first paragraph of the proof of Lemma 3.2) we conclude that there is a straight line j -arc joining a point of ∂N(p, ξ) to p. But this is impossible, since p is a spiral singularity. Thus indeed, by following the i-characteristic through z in either direction we will be led outside \overline{N(p, ε)}. From this it follows that there is a neighborhood S of z on ∂Δ(p, ε) such that no w in S can be joined to p by an i-arc lying in \overline{N(p, ε)}; that is, ∂N(p, ε)\A_{i}(ε) is open in ∂N(p, ε). From the connectedness of ∂N(p, ε) it now follows that A_{i}(ε) = ∂N(p, ε), which is to say that each point of ∂N(p, ε) is joined to p by characteristic arcs of both families lying inside this circle. That these arcs are not tangent to this circle follows from Lemma 3.1

We are now in a position to completely characterize type S singularities.

Theorem 3.1. If the HP-function θ has a singularity of type S at p, then in some neighborhood of p, θ(z) = arg(z − p) + α, where α is a constant.

Proof. In this proof we use U to denote the unit punctured disk, N′(0, 1). Without loss of generality we may assume that p = 0. Let δ_{1} be as in the conclusion of Lemma 3.3. By replacing θ with the HP-function θ(δ_{1}z) we may assume that δ_{1} = 1. It follows from Lemma 3.3 and the continuity of θ that e^{iθ(z)}/z must lie in one of the four open quadrants for all z in U\{0} and on a compact subset thereof on each concentric circle r∂U, 0 < r ≤ 1; to be specific, we shall assume that it is the second quadrant, so that the vectors e^{iθ(z)} and ie^{iθ(z)} point in the direction of movement towards 0 along the 1- and 2-characteristics, respectively. From the confluence of characteristics at 0 implied by Lemma 3.2, it follows that

(3.1)  (−1)^{i+1}D_{i}θ(z) ≥ 0  a.e. in U, i = 1, 2,

so that θ varies monotonically on characteristics in U. Because we have a singularity of type S at 0 it follows that θ tends monotonically to ∞ (−∞) as we move along 1-characteristics (2-characteristics) towards 0. The construction given just before the discussion of double Riemann nets in Section 1.4 shows that θ has a unique extension to an HP-function in C\{0}, so that for convenience we may regard it as being defined in the entire punctured plane. In light of that construction it follows that θ tends monotonically to −∞ (∞) as we move along 1-characteristics (2-characteristics) away from 0.

Let L_{i}(z) denote the distance along the i-characteristic from z to 0. From the convexity properties of the characteristics implied by (3.1) it follows that L_{i}(z) is nondecreasing as z moves along a j -characteristic away from 0. From Proposition 1.7 it follows that Di^{+}θ(z) ≤ 1/L_{j}(z). This means that L_{i}(z) is never infinite, since if it were infinite at some point, we would have straight line j -arcs, and by translating them along i-arcs down to 0 we would obtain straight line characteristic arcs terminating at 0, a contradiction. From this, in turn, it follows that |D_{i}θ(z)| = 1/L_{j}(z) a.e. on each i-characteristic, since otherwise a simple argument shows that the j -characteristics would not be confluent at 0. Since the function L_{j} is obviously continuous in the punctured plane, we must actually have that this last equation holds everywhere there; more precisely we have D_{i}θ(z) = (−1)^{i+1}/L_{j}(z), z ≠ 0. Thus the continuous branches of θ are C^{1} functions, and |D_{i}θ(z)| > 0 on C − {0}, i = 1, 2.

From the spiral nature of the characteristics, it follows that for any a in the punctured plane there are i-arcs C_{i}(a), i = 1, 2, which both join a to a point b = b(a) closer to 0, and whose only points in common are a and b. These arcs are, moreover, uniquely determined by these conditions. Let τ_{i}(a) denote Δθ(ab) along C_{i}(a), so that τ_{1}(a) > 0 and τ_{2}(a) < 0. Simple geometry implies that

(3.2)  τ_{1}(a) − τ_{2}(a) = 2π.

If we replace a by a point a′ near it on the 2-characteristic through it, then b will be replaced by b′ = b(a′). Consideration of the characteristic quadrilateral with vertices a, a′, b, b′ shows that τ_{1}(a′) = τ_{1}(a), which in light of (3.2) implies that τ_{2}(a) = τ_{2}(a′), also. Similarly, these functions are unchanged by small movements along 1-characteristics. This means that τ_{1} and τ_{2} are constants. In other words if, starting from any point a in the punctured plane, we move along the i-characteristic towards 0 in such a way that θ changes by τ_{i} we come to the same point independently of whether i is 1 or 2.

For a given characteristic C and two points a, b ∈ C we let Δ_{C}θ(a, b) denote the change in θ as one moves along C from a to b, i.e., Δ_{C}θ(a, b) = θ̄(b) − θ̄(a), where θ̄ denotes any branch of θ which is continuous on C. (This is the same “Δθ” notation we have been using up to now, with the subscript indicating the characteristic added to avoid possible confusion.) Let F_{i} be the i-characteristic through 1 ∈ C. For i = 1, 2 we define a mapping of R^{2} onto C\{0} as follows. For (t_{1}, t_{2}) ∈ R^{2} , let q be the point on F_{i} for which Δ_{F_{i}}θ(1q) = τ_{i}t_{i}/π and let z = ζ_{i}(t_{1}, t_{2}) be the point on the j -characteristic C through q for which Δ_{C}θ(qz) = τ_{j}t_{j}/π. It follows immediately from the defining property of HP-nets that if ζ_{1}(t_{1}, t_{2}) = ζ_{2}(t_{1}, t_{2}), then there is a δ > 0 such that ζ_{1}(t′_{1}, t_{2}) = ζ_{2}(t′_{1}, t_{2}) and ζ_{1}(t_{1}, t′_{2}) = ζ_{2}(t_{1}, t′_{2}) for all t′_{1} , t′_{2} such that |t′_{1} − t_{1}| < δ and |t′_{2} − t_{2}| < δ. Since ζ_{1}(0, 0) = 1 = ζ_{2}(0, 0), the continuity of ζ_{1} and ζ_{2} then implies that ζ_{1}(t_{1}, t_{2}) = ζ_{2}(t_{1}, t_{2}) for all t_{1}, t_{2} ∈ R; their common value will be denoted by ζ(t_{1}, t_{2}). Let (t_{1}, t_{2}) ∈ R^{2} and let q, r ∈ F_{1} be such that

Δ_{F_{1}}θ(1q) = \frac{τ_{1}t_{1}}{π}  and  Δ_{F_{1}}θ(1r) = \frac{τ_{1}(t_{1} + π)}{π} = \frac{τ_{1}t_{1}}{π} + τ_{1}.

Then by what was said in the preceding paragraph Δ_{C}θ(qr) = τ_{2} , where C is the 2-characteristic through q. From this it follows immediately that ζ_{1}(t_{1} + π, t_{2}) = ζ_{1}(t_{1}, t_{2} + π), that is, that

(3.3)  ζ(t_{1} + π, t_{2}) = ζ(t_{1}, t_{2} + π)  for all t_{1}, t_{2} ∈ R.

One easily calculates that

\frac{∂ζ}{∂t_{k}} = i^{k−1} \frac{τ_{k}}{π} \frac{e^{iθ(ζ)}}{D_{k}θ(ζ)}, k = 1, 2,

(using the fact that ζ(t_{1}, t_{2}) = ζ_{l}(t_{1}, t_{2}), where {k, l} = {1, 2}), so that since continuous branches of θ are C^{1} functions, ζ is also of class C^{1} . If we define

(3.4)  ω(t_{1}, t_{2}) = \frac{τ_{1}t_{1} + τ_{2}t_{2}}{π} + θ(1),

then it is clear that

(3.5)  arg\{\frac{∂ζ}{∂t_{1}} \} = ω(t_{1,}t_{2})  and  arg\{\frac{∂ζ}{∂t_{2}} \} = ω(t_{1,}t_{2}) + \frac{π}{2} .

We also point out that for n ∈ Z, T ∈ R, i = 1, 2,

(3.6)  ζ({(t_{1}, t_{2}) : nπ ≤ t_{i} < (n + 1)π, t_{j} ≥ T }) ⊃ N′(0, δ),

for some δ > 0.

For each z ∈ U let p_{i}(z) be the point on the i-characteristic through z for which Δθ(zp_{i}(z)) = (−1)^{i+1}2π, that is, the point on this characteristic which we arrive at by moving along it towards 0 from z in such a way that θ changes by 2π in the case i = 1 and by −2π in the case i = 2. We claim that |p_{i}(z)/z|, which is clearly less than 1 for all z in U, is actually bounded away from 1 there. If it were not, then there would be a sequence {z_{k}} of points in U tending to 0 and such that |p_{i}(z_{k})/z_{k}| approaches 1. Application of the compactness principle (Proposition 1.9) to the family of nets corresponding to the HP-functions θ(|z_{k}|z) would then yield a subsequence converging to a net ℋ in the entire punctured plane for which ∂U is an i-arc. But then it is clear that ℋ is the degenerate spiral net 𝒮_{0,π/2} . This in turn means that the original net would have to have i-arcs arbitrarily close to a full circle centered at 0, which contradicts the fact that |τ_{i}| < 2π. Thus indeed, the |p_{i}(z)/z| are bounded away from 1 in U. In terms of our characteristic coordinates this means that |ζ(t_{1}, t_{2})| decreases exponentially as either of the t_{i} tends to ∞; more precisely it means that there exist A, K > 0, such that

(3.7)  |ζ(t_{1}, t_{2})| < Ae^{−K(t_{1}+t_{2})},  for t_{1}, t_{2} ≥ 0.

In light of (3.5) and (3.6) we must show that if u(t_{1}, t_{2}) = ζ(t_{1}, t_{2}) · e^{iω}, v(t_{1}, t_{2}) = ζ(t_{1}, t_{2}) · ie^{iω}, then u/|ζ|, v/|ζ| are constant in some half-strip nπ ≤ t_{i} < (n + 1)π, t_{j} ≥ T. (Here again we are using the dot to denote inner product.) In terms of the real and imaginary parts x and y of ζ, the equations (3.5) may be written as (see equations (1.8))

cos ωy_{t_{1}} − sin ωx_{t_{1}} = 0  and  sin ωy_{t_{2}} + cos ωx_{t_{2}} = 0,

which in turn imply

(3.8)  v_{t_{1}} = −\frac{τ_{1}}{π} u  and  u_{t_{2}} = \frac{τ_{2}}{π} v.

For w = u or v we let w̄(s, t) stand for w(\frac{1}{2}(t + s), \frac{1}{2}(t − s)), so that by (3.2), (3.3) and (3.4), ū(s, t) and v̄(s, t) are periodic with period 2π in s.

Let γ^{2} = −τ_{1}τ_{2}/π^{2} , γ > 0. Since 0 < τ_{1}, −τ_{2} < 2π and τ_{1} − τ_{2} = 2π, it follows that 0 < γ ≤ 1. If we write equations (3.8) in integral form and take the bound (3.7) into account, we see that both u and v satisfy the equation

w(t_{1}, t_{2}) = γ^{2} ∫_{t_{1}}^{∞} ∫_{t_{2}}^{∞} w(s_{1}, s_{2}) ds_{2} ds_{1}.

A standard argument based on this equation, the bound (3.7), and the continuity of u and v shows that they are both C^{∞} -functions on R^{2} . The integral equation written in differential form is simply the well-known telegraph equation

w_{t_{1}t_{2}} − γ^{2}w = 0,

so that in terms of the variables s and t, ū and v̄ satisfy

w_{tt} − w_{ss} = γ^{2}w,

are periodic with period 2π in s and tend uniformly to 0 as t → ∞. It is easily established by separation of variables that the solutions of the telegraph equation for which w̄(s + 2π, t) = w̄(s, t), with smooth periodic Cauchy data on t = 0 are given in the upper half-plane t ≥ 0 by

w̄(s, t) = ∑_{k=0}^{∞} F_{k}(s, t),

where

F_{0}(s, t) = A_{0}e^{γt} + B_{0}e^{−γt}

and for k ≥ 1

F_{k}(s, t) = (A_{k} cos ks + B_{k} sin ks) cos((k^{2} − γ^{2})^{1/2}t)
+ (C_{k} cos ks + D_{k} sin ks) sin((k^{2} − γ^{2})^{1/2}t),

with the exception that when γ = 1

F_{1}(s, t) = A_{1} cos s + B_{1} sin s + (C_{1} cos s + D_{1} sin s)t.

When γ < 1 the uniform exponential decay of w̄(s, t) to 0 as t → ∞ implies that there is a K > 0 such that for t ≥ K, k ≥ 1,

|A_{k} cos((k^{2} − γ^{2})^{1/2}t) + C_{k} sin((k^{2} − γ^{2})^{1/2}t)| = |\frac{1}{π} ∫_{0}^{2π} w̄(s, t) cos ks ds| ≤ ε.

This clearly implies that A_{k} and C_{k} are 0 for k ≥ 1. Similarly, B_{k} and D_{k} are 0 for k ≥ 1. The case γ = 1 is handled in a similar way. Thus ū and v̄ are of the form A_{0}e^{γt} + B_{0}e^{−γt}. But since ū(s, t) → 0 as t → ∞, we must have A_{0} = 0, so that ū(s, t) = B_{0}e^{−γt}. Similarly, v̄(s, t) = B′_{0}e^{−γt}, for some other constant B′_{0} . But then |ζ| = (B_{0}^{2} +B′_{0}^{2})^{1/2}e^{−γt}, so that ū/|ζ| = B_{0}/(B_{0}^{2} +B′_{0}^{2})^{1/2} and v̄/|ζ| = B′_{0}/(B_{0}^{2} +B′_{0}^{2})^{1/2} are constants in the upper half-plane, and therefore u/|ζ| and v/|ζ| are indeed constant in some half-strip nπ ≤ t_{i} < (n + 1)π, t_{j} ≥ T.

Theorem 3.2. If ℋ ∈ HP^{∗}(C) has a spiral or degenerate spiral singularity, then ℋ is one of the 𝒮_{p,α}.

Proof. Let ℋ as in the hypothesis have a spiral singularity at p and let q be another singularity of ℋ at minimum distance from p. The convexity of the characteristics emanating from p shows that q is a Riemann singularity. But then Theorem 2.4 shows that there is, in fact, no singularity at q. Thus p is the only singularity of ℋ . The conclusion follows from Theorem 3.3 together with the uniqueness of the nets resulting from the construction process described just before the discussion of double Riemann nets in Section 1.4. The case of a degenerate spiral singularity is handled similarly.

Theorem 3.2 together with Theorem 2.5 gives us the following complete description of HP-nets with a single singularity.

Theorem 3.3. If ℋ is an HP-net regular in C except for a single singularity at p, then ℋ is an ℛ_{p,α,β} or an 𝒮_{p,α}.

4. More than one singularity

We begin with the following

Lemma 4.1. Let D be a Jordan domain with C^{1} boundary. Let θ be an HP^{∗} -function on D with singularity set A. Let R denote the set of nondegenerate Riemann singularities of θ. Then

(4.1)  ∑_{a∈R} dist(a, ∂D) ≤ \frac{2λ(∂D)}{π},

and

(4.2)  ∑_{a∈A\R} dist(a, ∂D) ≤ \frac{λ(∂D)}{π}.

Proof. We begin with the following observation. Let Q be a characteristic quadrilateral of θ in D\A one of whose j -sides is C = ab, and let E ⊂ Q be a simple arc joining the i-sides of Q. Assume, furthermore, that C is everywhere convex toward the inside of Q. Then

(4.3)  λ(E) ≥ |Δθ(ab)| min{|p − e| : p ∈ C, e ∈ E}.

To see this we recall that if p ∈ C is a point at which D_{i}θ exists and z = z(s), 0 ≤ s ≤ σ, is the arc length parametrization of the j -arc F_{C} joining p to the other j -side of Q, then the (unsigned) radius of curvature R(s) = 1/|D_{j}θ(z(s))| is differentiable and satisfies R′(s) = 1. Since D_{j}θ(p) exists for almost all p ∈ C and the length of the i-arc joining p to a point e ∈ E in Q\E is at least |p − e|, a simple calculus argument shows that (4.3) indeed holds.

It is clearly sufficient to show that if D′ is any smoothly bounded subdomain of D for which A ∩ ∂D′ = ∅, then

(4.4) ∑_{a∈R∩D′} dist(a, ∂D′) ≤ \frac{2λ(∂D′)}{π} and ∑_{a∈(A\R)∩D′} dist(a, ∂D′) ≤ \frac{λ(∂D′)}{π}.

It follows from Theorems 2.3 and 2.4 that each nonbounding characteristic of the i-fan F_{a,i} of a ∈ R has a unique subarc which joins a to some point ∂D′. Denote the set of all such points of ∂D′ by E_{a,i}. It is clear that for each i all of the sets E_{a,i} are disjoint. Let ε > 0 and let G be an open j -arc in N(a, ε) which joins the bounding characteristics of F_{a,i}. Then each point p ∈ G is contained in an open subarc C of G which is the side of a characteristic quadrilateral in D\A for which the other j -side lies outside of \overline{D′}. By the initial observation the length of the part of E_{a,i} in this quadrilateral is at least δ(dist(a, ∂D′) − ε), where δ is the change in θ on C. From this it follows that

λ(E_{a,i}) ≥ γ_{a,i} dist(a, ∂D′),

γ_{a,i} being the i-fan angle at a. Thus

∑_{i=1}^{2} ∑_{a∈R∩D′} γ_{a,i} dist(a, ∂D′) ≤ 2λ(∂D′),

so that the first bound of (4.4) follows upon taking into account that γ_{a,1} + γ_{a,2} = π.

The second inequality of (4.4) follows in essentially the same way. For a true spiral singularity a, by Theorem 3.1, θ coincides with σ_{a,α} for some α. As explained at the end of the discussion of spiral nets in Section 1.4, there are k -arcs C_{k}, k = 1, 2, contained in N(a, ε) along which θ has change β_{α,k} > 0 with β_{a,1} + β_{a,2} = 2π. If we replace the i-fan used above in the case of Riemann singularities with the set of j -arcs issuing from C_{i} and moving away from a, the argument proceeds just as in that case and we obtain that

λ(E_{a,i}) ≥ β_{a,i} dist(a, ∂D′).

The same holds in the case of degenerate spiral singularities, with the difference that one of the β_{a,k} vanishes. In any event, the second bound of (4.4) now follows immediately.

Corollary 4.1. Let D be a Jordan domain with C^{1} boundary. Let θ be an HP^{∗} -function on D with singularity set S. Then

∑_{a∈S} dist(a, ∂D) ≤ \frac{3λ(∂D)}{π}.

Corollary 4.2. An HP^{∗} -function θ on C can have at most four singularities.

Proof. Let p_{1}, …, p_{n} be distinct singularities of θ. If any of them is a spiral or degenerate spiral singularity then, by Theorem 3.2, n = 1. Thus we assume that they are all nondegenerate Riemann singularities. If N(0, R) is a disk containing all of these singularities and such that θ is regular on ∂N(0, R), then by Lemma 4.1

∑_{k=1}^{n}(R − |p_{k}|) ≤ 4R,

from which the desired conclusion follows upon letting R → ∞.

For each p ∈ D, a very similar argument shows that there is a neighborhood U = U(D, p) of p such that any θ ∈ HP^{∗}(D) can have at most four singularities in U. This, together with the compactness principle for HP(D) (Proposition 1.9) and a straightforward diagonal argument, gives the following

Corollary 4.3 (Compactness principle for HP^{∗}(D)). For any sequence {θ_{n}} of functions in HP^{∗}(D) there exists a subsequence {θ_{n′}} and a θ ∈ HP^{∗}(D) such that e^{2iθ_{n′}} converges to e^{2iθ} in the topology of uniform convergence on compact subsets of D\A, where A is the singularity set of θ.

The remainder of this section is devoted to proving the following

Theorem 4.1. An HP^{∗} -net in all of C can have at most two singularities. If it has no singularities, then the corresponding θ is a constant. If it has one singularity then it is either one of the Riemann nets ℛ_{p,α,β} or one of the spiral nets 𝒮_{p,α}. If it has two singularities then it is either one of the double Riemann nets 𝒟_{p,q,α,β,γ} or one of the degenerate double Riemann nets ℱ_{p,q,σ,β,γ}.

The case of zero or one singularity is covered by Proposition 1.8 and Theorem 3.3. In light of Theorem 3.2 any HP*-net in C which has a spiral or a degenerate spiral singularity is an 𝒮_{p,α}. Thus, for the remainder of this section we shall assume that ℋ is an HP-net which is regular in all of C except for distinct nondegenerate Riemann singularities p_{1}, p_{2}, …, p_{n}, 2 ≤ n ≤ 4. We shall show that the corresponding net is one of the double Riemann nets 𝒟_{p,q,α,β,γ} or one of the degenerate double Riemann nets ℱ_{p,q,σ,β,γ} described in Section 1.4. We begin with a number of preliminary observations and the introduction of some further notation and terminology that will facilitate the discussion.

It is easy to see that no singularity can be joined to itself by a characteristic. Indeed, if C is an i-characteristic which both begins and ends at one of the singularities p, then what we know about the i-fan of p implies that there is a subarc E of C whose endpoints are also joined by a j -arc E′ and such that the domain between E and E′ contains no singularities, an obvious impossibility. Thus, each characteristic either emanates from some singularity and has infinite length, extends from one singularity to another in finite length or extends infinitely in both directions. If p is a singularity, the fan of i-characteristics emanating from p will be denoted by F_{i}(p). In addition, R_{i}(p) and L_{i}(p) will denote the bounding characteristics of F_{i}(p) on the right and left sides of this fan (with respect to movement away from p). For each pair (p, i), where p is a singularity and the angle of F_{i}(p) is positive, we choose a fixed nonbounding characteristic M_{i}(p) of F_{i}(p) and let z(s), 0 < s < L, be the arc length parametrization of M_{i}(p). For all sufficiently small s there is an open j -arc C_{j}(p, s) containing z(s) and joining a point of R_{i}(p) to a point of L_{i}(p). This arc is everywhere concave towards p. Because of this, in light of Theorems 2.3 and 2.4, all characteristics of F_{i}(p), with the possible exception of R_{i}(p) and L_{i}(p), can be extended indefinitely away from p (without terminating at any other singularity), so that L = ∞ and C_{j}(p, s) is well defined for all s > 0. (The arbitrariness of the choice of M_{i}(p) will be of no consequence.)

Lemma 4.2. Let p be a singularity of an HP-function θ on C which has only nondegenerate Riemann singularities. Let C be an i-arc of θ with arc length parametrization z(s), 0 < s < L < ∞. Then there exist 0 = s_{0} < s_{1} < ⋯ < s_{m} = L such that for 1 ≤ i ≤ m − 2, z((s_{i,}s_{i+1})) is either a straight line segment or C_{j}(a, s) for some singularity a and some s > 0. For i = 0 or m−1, z((s_{i,}s_{i+1})) is either a straight line segment or a subarc of some such C_{j}(a, s).

Proof. The only way this could fail to be so would be for there to be infinitely many disjoint arcs of the form C_{j}(a, s) on C. But, since the associated fan angles have at most four different positive values, this is inconsistent with the fact that θ(z) is continuous on C and possesses limits as z approaches the endpoints of C.

The proof that the only possible nets are the (nondegenerate and degenerate) double Riemann nets is broken down into two main cases according to whether or not some pair of (distinct) singularities are joined by a characteristic; each of these cases is further broken down into subcases. In each instance the proof reduces to showing that there are only two singularities and that a certain characteristic arc is a straight line segment.

Case I. There are singularities p, q joined by a characteristic i-arc C. We begin by noting the following simple consequences of this hypothesis.

(1) There is a neighborhood N of C such that θ is regular in N − {p, q}. This is obvious from the definition of isolated singularity and the regularity

of θ along C.

(2) C is the only i-arc joining p to q.

Assume, to the contrary, that there is another one C′. Then from what we know about the structure of Riemann singularities, C and C′ both belong to both F_{i}(p) and F_{i}(q). But then the orthogonal characteristics connecting C and C′ inside these fans must have nonvanishing curvature, and have their concave side towards both p and q, which is impossible.

(3) C ∈ {R_{i}(p), L_{i}(p)} ∩ {R_{i}(q), L_{i}(q)}.

If C were neither R_{i}(p) nor L_{i}(p), then from the structure of Riemann singularities it follows that there are j -characteristic arcs forming an angle of π at q and which are strictly concave towards p. But this contradicts Theorem 2.4.

(4) C is R_{i}(p) and R_{i}(q) or L_{i}(p) and L_{i}(q).

If either of the fans F_{i}(p) or F_{i}(q) is made up of a single characteristic then there is nothing to prove, so we assume that this is not the case. Say that C is R_{i}(p). If it were L_{i}(q), then it is easy to see that there would be two distinct i-characteristics joining p and q, which contradicts point (1) above.

We now begin the analysis of Case I. For the sake of definiteness we assume C is R_{i}(p) and R_{i}(q). We also assume for the time being that angles β, γ of the fans F_{i}(p) and F_{i}(q) satisfy 0 < β, γ < π; we will discuss the degenerate cases separately. The accompanying Figure 3 has been included to facilitate understanding the following discussion. We shall show that the net is ℱ_{p,q,+,β,γ}, so that this figure is to be compared to Figure 2. Let e ∈ {p, q}. We consider that C_{j}(e, s) is oriented so that movement in the positive direction along it coincides

Figure 3.

with increasing θ; the initial and terminal points of its closure will be denoted by a_{e}(s) and b_{e}(s); for completeness we set a_{e}(0) = b_{e}(0) = e. For small s, a_{e}(s) lies on the curve C. Let {e, f} = {p, q}. We follow the movement of C_{j}(e, s) as s varies from 0 to ∞. On an initial interval [s_{0,e}, s_{1,e}], s_{0,e} = 0, a_{e}(s) traces out C from e to f, while b_{e}(s) traces out an arc of L_{i}(e). Next, on an interval [s_{1,e}, s_{2,e}], a_{e}(s) is stable at f while b_{e}(s) traces out a j -arc which is concave towards f. Then, on an interval [s_{2,e}, s_{3,e}], a_{e}(s) runs along the arc b_{f}([s_{0,f}, s_{1,f}]). Continuing in this manner we see that there are two increasing sequences {s_{k,e}}, e = p, q, such that on [s_{k+2,e}, s_{k+3,e}], a_{e}(s) runs along the arc b_{f}([s_{k,f}, s_{k+1,f}]) of L_{i}(f). The arcs C_{j}(e, s), s ∈ [s_{k+2,e}, s_{k+3,e}] and C_{j}(f, s), s ∈ [s_{k,f}, s_{k+1,f}] lie on opposite sides of L_{i}(f). In addition, they are both concave towards f. From this, together with Theorem 2.3 it follows that L_{i}(p) and L_{i}(q) are both infinite (that is, as we move out from p and q along them we never encounter singularities.

For k ≥ 1, let Q_{k}(e) denote the (closed) characteristic quadrilateral with vertices b_{e}(s_{k−1,e}), a_{e}(s_{k−1,e}), a_{e}(s_{k,e}) and b_{e}(s_{k,e}). These are all bona fide characteristic quadrilaterals with the exception of Q_{1}(e), Q_{2}(e) which have degenerate sides at ee and ff, respectively. The fact that there are no singularities on any of the C_{j}(e, s), s > 0, nor on the L_{i}(e) means that the only singularities in any of the Q_{k}(e) are the points p and q. It is easy to see that |b_{e}(s) − a_{e}(s)| ≥ Kλ(C_{j}(p, s)) → ∞ so that the union of the Q_{k} is the entire plane. Thus indeed the only singularities are p and q. Finally, since none of the points of C is joined to a singularity by a j -characteristic, C is a straight line segment. Comparison with the construction of the degenerate double Riemann nets now shows that this net is indeed ℱ_{p,q,+,β,γ}.

We still have to discuss those degenerate situations in which the angles of one or both of the fans F_{i}(p) and F_{i}(q) is 0 or π. By interchanging p and q, if necessary, it is sufficient to consider the following possibilities:

(A) 0 < β < π, γ = 0. (B) 0 < β < π, γ = π. (C) β = γ = π. (D) β = 0, γ = π. (E) β = γ = 0.

Cases A–D. Cases A–D are very similar to the case in which 0 < β, γ < π, which we have just treated, the only difference being that some of the quadrilaterals Q_{k}(p), Q_{k}(q) degenerate into characteristic arcs or do not even appear, and the proof proceeds with minor alterations, the details of which we leave to the reader. In Case A none of the Q_{k}(q) appear, because the fact that γ = 0 causes the C_{j}(q, s), in effect, to degenerate to a point. Nonetheless, the Q_{k}(p) cover the plane, and allow us to reach the desired conclusion as before. In Case B the degeneracy of C_{i}(q, s) makes Q_{2}(p), Q_{4}(q), Q_{6}(p), Q_{8}(p), … degenerate into j -arcs, but again the remaining quadrilaterals cover the plane in the necessary fashion. In Case C it is all the even numbered quadrilaterals that degenerate into j -arcs because of the degeneracy of both C_{i}(p, s) and C_{i}(q, s), but once again the remaining quadrilaterals cover the plane in the necessary manner. Finally, in Case D (see Figure 4) all of the Q_{k}(p) degenerate (more precisely, Q_{2}(p) degenerates to the point q and all the other Q_{k}(p) degenerate to arcs) because the C_{j}(p, s) and the C_{i}(q, s) degenerate to points; again the Q_{k}(q) cover the plane. In all four of these cases the only singularities are p and q and the i-arc C is a straight line, so that the net is seen to be ℱ_{p,q,+,β,γ}.

Case E. The treatment of this case differs substantially from that of the other four, since none of the quadrilaterals actually appears because of the degeneracy of both C_{j}(p, s) and C_{j}(q, s). Let M_{e} be any fixed nonbounding characteristic of F_{j}(e), and let z_{e}(s), 0 < s < ∞ be an arc length parametrization of M_{e}. (The characteristics M_{e} are infinite by Theorems 2.3 and 2.4 as pointed out above.) We apply the decomposition of Lemma 4.2 to M_{e}. The first subarc must be a straight line segment, since otherwise it would have to be a subarc of a C_{j}(f, s) or C_{j}(e, s), which is inconsistent with the hypothesis that the angle of the j -fan at both e and f is 0. Let s_{1} > 0 be such that G = z_{e}((0, s_{1})) is a straight line segment. It follows that for small positive s, there is a single i-characteristic N_{s}, which is a simple closed curve consisting of semicircles of radius s (which are C_{j}(p, s′), C_{j}(q, s′) for appropriate values of s′) joined by translates of C. Let σ be a fixed such s. Let w(t), 0 ≤ t ≤ L, be an arc length parametrization of N_{σ} traversed once starting at and ending on G. Let G_{t} denote the associated translates of G along N_{σ}. We claim that θ is regular on all of these G_{t}. If not, let σ′, where σ < σ′ < s_{1} , be the smallest number for which a singularity is encountered on a translate of z_{e}((0, σ′]). In the event that there is more than one such singularity we consider the one lying on the G_{t} with the smallest possible value of t ∈ (0, L). It is clear that for all s < σ′, there is an N_{s} as described above.

It is also clear that for this singularity the angle of the j -fan is π, since it lies in the interior of one of the straight j -arcs G_{t}. But this means that points z_{e}(s), for s < σ′ near σ′ are joined by i-characteristics to points z_{e}(s), for s > σ′ near σ′, which is inconsistent with the existence of the simple closed i-characteristics N_{s}, of the kind described above, for s < σ′. Now assume that in the decomposition of M_{e} according to Lemma 4.2 we have a second interval, that is, z_{e}((0, s_{1})) is a straight line segment, but z_{e}((s_{1}, s_{2})) = C_{j}(c, τ), for some singularity c, and some τ > 0. But then the argument we used before will again show that there can be no singularities on the translates of z_{e}((0, s_{2})) around N_{σ} away from c, that is, in the direction of the side towards which z_{e}((s_{1}, s_{2})) is convex. But this contradicts the existence of the singularity c. Thus no such s_{1} exists, and the M_{e} are both straight lines on all of whose translates θ is regular. From this it follows the j -characteristic through each point of C is a straight line (infinite in both directions) so that C is itself straight. From the straightness of the M_{e} it follows that the net in question is ℱ_{p,q,+,0,0} = ℱ_{p,q,−,0,0} .

Case II. No two singularities are joined by a characteristic arc.

We assume that there are at least two singularities and that no two of them are joined by a characteristic. Let p be a singularity. It follows that the bounding characteristics of the fans F_{i}(p), i = 1, 2, are all free of singularities, so that when we apply the decomposition of Lemma 4.2 to each of them the initial piece is always a line segment. It is easy to see that all of these initial segments must be finite. Assume, to the contrary, that all the characteristics making up F_{i}(p) are rays. It is enough to show that all translates of R_{i}(p) along L_{j}(p) (and of L_{i}(p) along R_{j}(p)) are free of singularities, since then L_{j}(p) and R_{j}(p) will be rays also, from which the desired conclusion follows immediately. Say, to the contrary, that as we translate R_{i}(p) along L_{j}(p) we encounter a first singularity at q. It is clear that the angle of the j -fan at q must be 0. Then L_{i}(q) is a ray. But this cannot be because L_{i}(q) contains C_{i}(p, τ) for some τ, and these curves are nowhere straight. Thus, indeed, all of the initial segments are finite.

We examine translates of these initial segments; to be specific we consider the initial line segment S_{R} of R_{i}(p). It is clear that the initial straight line segment S_{L} of L_{i}(p) has the same length σ as S_{R}. Let {T, U} = {R, L}. Let z_{U}(s), 0 < s < ∞, be the arc length parametrization of U_{j}(p). Let s_{T} be the smallest s > 0 for which the translate S_{T} (s) of S_{T} along U_{j}(p) to z_{U}(s) has a singularity, where s_{T} = ∞ if no such s exists. If s_{T} is finite it is clear that the singularity encountered must be at distance \frac{1}{2}σ from U_{j}(p), and that the angle of its j -fan must be 0 with this j -fan consisting of a single characteristic extending back towards F_{i}(p) at distance \frac{1}{2}σ from U_{j}(p). From this it follows that we cannot have both s_{R} and s_{L} finite, since we would then have two singularities joined by a characteristic. We bear these preliminary comments in mind as we deal with the various subcases into which Case II is divided.

Figure 4.

Case II-1. There are no singularities with fan angles 0 and π.

It follows from the preceding comments that all four initial segments are of finite length and that all their translates are free of singularities. Consider the initial segments S_{R} and S_{L} on R_{i}(p) and L_{i}(p), respectively. In reference to the decomposition of Lemma 4.2, the piece of R_{i}(p) which follows the straight line segment S_{R} is of the form C_{i}(q, σ) for some singularity q. In light of the underlying assumption the angles of both fans at q are strictly between 0 and π, we can assume this arc curves away from the fan F_{i}(p) (that is, that as we move away from p along this piece of R_{i}(p), the argument of the tangent decreases), since if it had the opposite sign we could work with S_{L} and L_{i}(p) instead of S_{R} and R_{i}(p) and have it curve away from the fan. The remainder of the discussion of this case is very similar to that of the nondegenerate Case I. Indeed, let e be p or q. For 0 < s < ∞, we let E(e, s) be the open i-arc made up of the C_{i}(e, s), its endpoint c(s) on L_{j}(e) and the translate of S_{R} one of whose endpoints is c(s). Furthermore, we let E(e, 0) be the translate of S_{R} one of whose endpoints is e and which lies to the left of L_{j}(e) (when movement along this characteristic is away from e), so that E(e, s) is continuous for s ≥ 0. (Note that E(p, 0) = S_{R}.) The desired conclusion is obtained by following the movement of E(p, s) and E(q, s) as s varies from 0 to ∞. Let {e, f} = {p, q}. We consider that E(e, s) is oriented so that the positive direction along it coincides with that of nondecreasing θ and we denote the initial and terminal points of its closure by a_{e}(s) and b_{e}(s). We let s_{0,e} = 0. Then on an interval [s_{0,e}, s_{1,e}], b_{e}(s) traces out an arc of L_{j}(f), which ends at f. Then we define s_{k,e}, k ≥ 2, inductively by b_{e}(s_{e,k}) = a_{e}(s_{k−2,f}). Note that b_{e}(s) = f, for s ∈ [s_{1,e}, s_{2,e}]; on this interval the arcs E(e, s) are initial arcs of the characteristics of the i-fan at f.

For k ≥ 1, let Q_{k}(e) denote the (closed) characteristic quadrilateral with vertices b(s_{k−1,e}), a(s_{k−1,e}), a(s_{k,e}) and b(s_{k,e}). These are all bona fide characteristic quadrilaterals, except for Q_{2}(e) which has the degenerate side ff. The fact that there are no singularities on any of the E(e, s), s > 0, nor on the R_{j}(e), L_{j}(e) means that the only singularities in any of the Q_{k}(e) are the points p and q. It is easy to see that |b_{e}(s) − a_{e}(s)| ≥ Kλ(E(p, s)) → ∞ so that the union of the Q_{k} is the entire plane. Thus indeed the only singularities are p and q. Finally, it is clear that none of the points of c([s_{0,e}, s_{1,e}]) is joined to a singularity by an i-characteristic, so that this arc is a straight line segment. Comparison with the construction of the double Riemann nets in Section 1.4 now shows that in this case the net is 𝒟_{p,q,β,γ,α}, where β is the angle of F_{i}(q), γ is the angle of F_{i}(p) and α is the inclination of the initial segment of L_{j}(p). This finishes the discussion of Case II-1.

Case II-2. There is a singularity with fan angles 0 and π at p. This case is further divided into the following subcases, where the angles of F_{i}(p) and F_{j}(p) are 0 and π, respectively.

(A) All translates of the initial segments at p are free of singularities.

(B) The midpoint of some translate of an initial segment of L_{i}(p) = R_{i}(p) is a singularity of θ.

(C) The midpoint of some translate of the initial segments of L_{j}(p) and R_{j}(p) is a singularity of θ.

Case A. The following discussion is illustrated in Figure 5. As in Case II-1, all initial segments are of finite length and all their translates are free of singularities. The piece of R_{j}(p) which follows its initial line segment S_{R} is of the form C_{j}(q, σ), σ > 0, for some singularity q. We may assume that as one moves along R_{j}(p) away from p, θ decreases; if this is not the case, one works with L_{j}(p) instead of R_{j}(p). We assume that the angles of the fans at q are strictly between 0 and π; the minor modifications required if this is not so (i.e., the angle of F_{i}(q) is π) are left to the reader. By assumption R_{i}(p) = L_{i}(p) is free of singularities. Let l > 0 be the length of S_{R}. There are translates of S_{R} (which are all free of singularities) with endpoints on R_{i}(p) extending outward on both sides of R_{i}(p). Let z(s), s ≥ 0, be the arc length parametrization of R_{i}(p), and let E(s) denote the open line segment j -arc of length 2l and with midpoint z(s). We see that starting with E(0), E(s) moves towards q with one endpoint on L_{i}(q), and then on some interval [τ, τ′] has an endpoint at q while it rotates around q in the counterclockwise direction forming initial segments of the characteristics in F_{j}(q), until E(τ′) forms the initial segment of L_{j}(q). Thereafter it has one of its endpoints on R_{i}(q). We now define E′(s), s > τ′, to be the j -arc consisting of E(s), its endpoint on R_{i}(q) together with the C_{j}(q, t) which shares this endpoint. For s = τ′ we let E′(s) consist of E(τ′) alone (since the corresponding C_{j}(q, t) tends to the single point q as t tends to 0). It is clear that there are no singularities on any of the E′(s) since there are none on the C_{j}(q, t), on R_{i}(q) or on the E(s). It follows from the definition of the E′(s) that both of their endpoints are on L_{i}(q). This, together with the fact that L_{i}(q) is free of singularities, implies that

L_{i}(q) ∪ ( ⋃_{s>τ′} E′(s)) ∪ ( ⋃_{0≤s≤τ′} E(s)) ∪ A = C,

where A denotes the semidisk consisting of those translates of S_{R} which are the initial segments of the characteristics of F_{j}(p). But then the only singularities of θ are p and q, so that in particular there are no singularities on any of the translates of z((0, τ)), which means that this arc is a straight line segment. Hence, this net is 𝒟_{p,q,π,β,α}, where β is the angle of F_{j}(q), and α is the inclination of the initial segment of L_{i}(q).

Case B. It is easy to see that this case is identical to the situation which arises in the foregoing Case A when the angle of F_{i}(q) is π if we interchange the roles of p and q, and i and j.

Case C. To be specific, assume that the midpoint m of the translate of the initial segment S_{R} of R_{j}(p) along L_{i}(q) = R_{i}(q) is the singularity q. The line joining m to q is the initial arc of L_{i}(q) = R_{i}(q); by the defining assumption of Case II one encounters no singularities as one moves along R_{i}(q) away from q. Let z(s), s ≥ 0, be the arc length parametrization R_{i}(q), and let E(s) denote that translate of S_{R} whose midpoint is z(s). Let τ > 0 be such that E(τ) = S_{R}. We assign the (continuously varying) initial point a(s) of E(s) so that a(τ) = p. Then a(s) moves back along R_{i}(p) from a(0) to p, remains at p while E(s) rotates through an angle of π about p in the counterclockwise direction then moves along R_{i}(p) for s > τ′ = τ + |p − m|π. As s > τ′ increases a(s) traverses a translate of z((0, τ)), a semicircle C_{j}(q, t), another translate of z((0, τ)), a semicircle C_{j}(p, t′), and so on. From this it is easy to see that the E(s), s ≥ 0, together with R_{j}(p) (including p itself) make up all of C. But then once again the only singularities of θ are p and q, so that in particular none of the points of z((0, τ)) is joined to a singularity by a j -characteristic, and therefore this arc is a line segment. Hence, the net is 𝒟_{p,q,π,π,α}, where α is the inclination of the initial segment of R_{i}(q).

5. Boundary limits and constant principal strain mappings

We begin with the following

Lemma 5.1. Let θ be a nonconstant HP-function on a simply connected domain D. Then there is an open interval I such that for all M ∈ I the (M, 1)- mappings corresponding to θ are not one-to-one in D.

Proof. It follows from the hypothesis that there is a point z_{0} at which θ is differentiable and at which its gradient does not vanish. Assume for the moment that κ_{0} = D_{1}θ(z_{0}) ≠ 0. Without loss of generality we may also assume that θ(z_{0}) = 0. Let z(s), |s| < s_{0} , z(0) = z_{0} , be an arc length parametrization of an open arc of the 1-characteristic of θ through z_{0} , so that z′(s) = e^{τ(s)i} with τ(s) = θ(z(s)). Then τ(s) = κ_{0}s + o(s), as s → 0. Let f be an (M, 1)-mapping corresponding to the HP-function θ, and let w(s) = f(z(s)). Again without loss of generality we can assume that w′(0) > 0. It then follows from the compatibility equations (1.1) and the fact that f is an (M, 1)-mapping that

w′(s) = Me^{Mτ(s)i} = Me^{Mκ_{0}si+Mo(s)} = Me^{Mκ_{0}si}(1 + Mo(s)),

so that for |t| ≤ 4π/M|κ_{0}|

\begin{aligned}w(t) − w(0) & = ∫_{0}^{t} Me^{Mκ_{0}si}(1 + Mo(s)) ds = \frac{(e^{Mκ_{0}ti} − 1)}{κ_{0}i} + M^{2}o((\frac{4π}{Mκ_{0}} )^{2}) \\ & = \frac{(e^{Mκ_{0}ti} − 1)}{κ_{0}i} + o(1), as M → ∞. \end{aligned}

From this together with the fact that on 2-arcs f does not alter arc length and changes curvature by a factor of 1/M (as follows from the second equation in (1.1)) it is clear that for sufficiently large M, f is not one-to-one in D. If we had D_{2}θ(z_{0}) ≠ 0, the analogous procedure would give us noninvertible (1, M)- mappings f for all sufficiently large M; the desired mappings are then given by (1/M)f.

Definition 5.2. We say that θ_{0} is the nontangential limit of θ ∈ HP^{∗}(D) at p ∈ ∂D if there exists a ξ ∈ R such that given any ε > 0 there is a δ > 0 with the property that the set

U = \{p + z : | arg\{z/e^{iξ}\}| < \frac{1}{2}π − ε\} ∩ N′(p, δ)

consists entirely of regular points of θ in D and the limit as z tends to p of some single-valued branch of θ(z) in U is θ_{0} .

Theorem 5.1. Let D be a domain bounded by a C^{1} Jordan curve. If θ is an HP^{∗} -function on D, then θ has nontangential limits at almost all points of ∂D.

Proof. Let θ be as in the hypothesis and let S denote the set of singularities of θ. We shall use (M, 1)-mappings f for which θ_{f} = θ. To do so, however, we must work with a simply connected subdomain of D, which we obtain by removing from D small pieces which join the points of S to ∂D. Every point of ∂D is contained in an almost straight closed subarc C such that there is a circular arc C′ lying in D which joins the end points of C and contains no point of S. Obviously, it is enough to prove the theorem for the domain bounded by C ∪ C′, so that we shall assume that D has this form. Clearly, there is a bi-Lipschitz homeomorphism Λ of D onto the rectangle R with vertices −2, 2, 2 + 2i, −2 + 2i such that Λ(S) is contained in the rectangle \frac{1}{2}R and such that the sum of the imaginary parts of all of the points in Λ(S) is at most 1. That the latter is possible follows immediately from Corollary 4.1. Let z_{k} = x_{k} + y_{k}i be an enumeration of the points of Λ(S) with k starting at 1. Let ε > 0 and let q_{k,ε} be the real function on [−2, 2] whose graph consists of the segments [−2, x_{k} − ε/2^{k}], [x_{k} −ε/2^{k}, z_{k}], [z_{k}, x+ε/2^{k}], [x+ε/2^{k}, 2], and let q_{ε} be the sum of all of the q_{k,ε}. We have 0 ≤ q_{ε}(x) ≤ 1, x ∈ [−2, 2]. The domain R_{ε} bounded by the graph of q_{ε} and ∂R\[−2, 2] is a Jordan domain contained in R\Λ(S) whose boundary contains all of ∂R except for a set of 1-dimensional measure at most ε. Let δ > 0. Since Λ is bi-Lipschitz, for sufficiently small ε, D^{∗} = Λ^{−1}(R_{ε}) ⊂ D\S is a Jordan domain whose boundary is rectifiable and contains all of ∂D except for a set of 1-dimensional measure less than δ. Since δ > 0 is arbitrary it is sufficient to show that (any continuous branch of) θ on D^{∗} has nontangential boundary limits a.e. on ∂D^{∗}. (Implicit in this construction of D^{∗} was the assumption that S ≠ ∅; if S = ∅ we let D^{∗} = D.)

Figure 5.

For each positive M ≠ 1, let f_{M} be an (M, 1)-mapping of D^{∗} corresponding to θ. Let v(s), 0 ≤ s ≤ L, v(0) = v(L) be a positively oriented arc length parametrization of the simple closed curve ∂D^{∗}. Obviously, v is differentiable on a subset B of (0, L) of measure L. It follows from the manner in which D^{∗} was constructed that there is a K such that given any η > 0, for any two ξ_{1}, ξ_{2} ∈ (0, L), v(ξ_{1}) and v(ξ_{2}) can be joined in D^{∗} by an arc of length at most K|ξ_{2} − ξ_{1}| + η. (Note that the domain R_{ε} used to construct D^{∗} in the preceding paragraph has this property with K = 1.) Because of this and the fact that f_{M} is Lipschitz, f_{M} has a unique continuous extension to \overline{D^{∗}}. Clearly, ψ_{M}(s) = f_{M}(v(s)) is Lipschitz continuous with constant K max{M, 1/M}, so that it is differentiable on a set A_{M} ⊂ (0, L) of measure L. Let P denote the set of positive rationals different from 1, and let A = ⋂_{M∈P} A_{M}. Let ξ ∈ B ∩A. It is enough to show that the nontangential limit of θ exists at v(ξ) since λ(∂D\v(B ∩ A)) = 0. Without loss of generality we may make the normalizing assumption that v(ξ) = 0 and arg v′(ξ) = 0. By appropriately choosing the constants of integration we may also stipulate that ψ_{M}(ξ) = 0.

Let H denote the open upper half-plane. All (M, 1)-mappings in H have a unique continuous extension to H which is Lipschitz continuous with constant max{M, 1/M}, and in dealing with such mappings we implicitly consider that they have been so extended. Although θ_{t}(z) = θ(tz) is not defined in all of H, for any compact X ⊂ H, θ_{t} is defined in X for all sufficiently small t, so that it is meaningful to talk about local uniform convergence in H of θ_{t} as t → 0. Indeed, we will have established the desired conclusion if we can show that θ_{t} converges in H to a constant in this sense. To do so we show first that if for some {t_{k}} approaching 0, θ_{t_{k}} tends locally uniformly to χ, then χ must be a constant χ_{0} which satisfies

(5.2)  4 cos^{2} χ_{0} + sin^{2} χ_{0} = (ψ′_{2}(ξ))^{2}.

Let {t_{k}} be such a sequence. By replacing it with a subsequence, if necessary, we may assume that for each M ∈ P the (M, 1)-mappings f_{M}(t_{k}z)/t_{k} converge locally uniformly to an (M, 1)-mapping g_{M} corresponding to χ. Since ξ ∈ B ∩A, it follows that the g_{M} are linear on ∂H = R. Assume that χ were not a constant. By Lemma 5.1 there is a μ ∈ P such that g_{μ} is not one-to-one on H. For convenience we assume that μ > 1; the opposite case is dealt with in the same way apart from minor notational differences. By general injectivity criteria for quasi-isometries developed by John [J3],

τ = inf{M ∈ P ∩ (1, ∞) : g_{M} is noninjective on H} > 1.

Let {M_{k}} be a sequence in P ∩ (τ, τ + 1) approaching τ and such that g_{M_{k}} is noninjective in H. That is, for each k there are distinct points a_{k}, b_{k} ∈ H for which g_{M_{k}}(a_{k}) = g_{M_{k}}(b_{k}).

It also follows from results of John [J3] that if g is an (M, 1)-mapping in N(a, r), then g is one-to-one in N(a, r/M). Thus Im a_{k}, Im b_{k} ≤ (2+τ)|b_{k} −a_{k}|, since if, for example, Im b_{k} > (2 + τ)|b_{k} − a_{k}|, then g_{M_{k}} would be one-to-one in N(b_{k}, (2 + τ)|b_{k} − a_{k}|/(1 + τ)), which contains both a_{k} and b_{k}. Let a′_{k} = (a_{k} − Re a_{k})/|b_{k} − a_{k}| and b′_{k} = (b_{k} − Re a_{k})/|b_{k} − a_{k}|. Then {a′_{k}} and {b′_{k}} are bounded. By replacing {M_{k}} by an appropriate subsequence we may assume that a′_{k} and b′_{k} converge to a and b, respectively. Clearly, |b − a| = 1. Let

h_{k}(z) = (g_{M_{k}}(|b_{k} − a_{k}|z + Re a_{k}) − g_{M_{k}}(Re a_{k}))/|b_{k} − a_{k}|.

Then h_{k} is an (M_{k}, 1)-mapping of H associated with the HP-function χ_{k}(z) = χ(|b_{k} − a_{k}|z + Re a_{k}) for which h_{k}(a′_{k}) = h_{k}(b′_{k}) and h_{k}(0) = 0. Again, by replacing {M_{k}} with an appropriate subsequence we may assume that there is an HP-function χ′ on H such that e^{2iχ_{k}} converges to e^{2iχ′} locally uniformly on H and that the h_{k} converge to a (τ, 1)-mapping h of H associated with χ′; the convergence of the h_{k} is locally uniform on H. But since for M ∈ (1, τ), (M, 1)-mappings associated with χ are injective, so are those associated with χ_{k} and therefore also those associated with χ′. If l_{M} denotes the (M, 1)-mapping associated with χ′ for which l_{M}(i) = h(i) and for which θ_{l_{M}} (i) = θ_{h}(i) and ϕ_{l_{M}} (i) = ϕ_{h}(i), then l_{M} → h as M → τ uniformly on compact subsets of H. Since the l_{M} are injective on H for M ∈ (1, τ), so is h. On the other hand, h(a) = h(b), since h_{k}(a′_{k}) = h_{k}(b′_{k}). Thus, a, b ∈ ∂H. However, since each h_{k} is linear on ∂H, so is h. Since a ≠ b, this means that h is a constant on ∂H. But this cannot be since it would imply that the image under h of the strip 0 < Im z < 1, which obviously has infinite area, is contained in a disk of radius τ. Our assumption that χ is not a constant has therefore led to a contradiction, so that χ is indeed a constant χ_{0} . That χ_{0} satisfies equation (6.2), follows from the fact that g_{2} is a linear (2, 1)-mapping of H.

Finally, we show that the θ_{t} do in fact converge as t tends to 0. If they did not, the compactness principle (Proposition 1.9) together with the preceding paragraphs, would imply that there are two sequences {t_{k}} and {t′_{k}} for which the corresponding sequences {θ_{t_{k}}} and {θ_{t′_{k}}} converge to constants χ_{1} < χ_{2} , both of which satisfy equation (5.2) above. Let χ_{3} ∈ (χ_{1}, χ_{2}) be a number which does not satisfy that equation. Let z_{0} ∈ H. Since θ(t_{k}z_{0}) and θ(t′_{k}z_{0}) tend to χ_{1} and χ_{2} , respectively, it follows from the intermediate value theorem that for all sufficiently large k there is an s_{k} between t_{k} and t′_{k} for which θ(s_{k}z_{0}) = χ_{3} . But then by the compactness principle {θ_{s_{k}}} has a subsequence which converges to an HP-function χ for which χ(z_{0}) = χ_{3} . This contradicts what was established in the preceding paragraph.

We now relate the class cps*(D, m_{1}, m_{2}) to HP^{∗}(D). We define the dilatation μ(p) = μ_{H}(p) of an HP-net H (for which the two families of characteristics have been numbered) as follows. If p is a Riemann singularity with k -fan angles α_{k}, k = 1, 2, then μ(p) = α_{2}/α_{1} , when α_{1}, α_{2} ≠ 0, \frac{1}{2}π, π, and is undefined otherwise. If p is a spiral singularity and α_{1} is the positive acute angle between the 1-characteristics and the rays emanating from p, then μ(p) = cot^{2} α_{1} , when α_{1} ≠ 0, \frac{1}{2}π, π, and is undefined otherwise. There is an (m_{1}, m_{2})-mapping f in a neighborhood of an isolated singularity if and only if μ(p) = m_{1}/m_{2} . Indeed, the relevant calculations for the spiral nets 𝒮_{p,α} were made in (i) of Section 1.4. The case of Riemann singularities follows immediately from the observation that if θ has a Riemann singularity at p with fan angles α_{1} and α_{2} , then a necessary and sufficient condition that there be a one-to-one single-valued (m_{1}, m_{2})-mapping f in a neighborhood of p is that the images of these fans have angles which sum to π. Since these angles are given by (m_{j}/m_{i})α_{i} the desired conclusion follows immediately. A standard argument then shows that in order for an HP^{∗} -net in a simply connected domain D to correspond to some (m_{1}, m_{2})-mapping with the same set of singularities, it is necessary and sufficient that for all singularities p of ℋ , μ_{ℋ} (p) = m_{1}/m_{2} . Note also that from this we have the obvious corollary to Theorem 4.1: any f ∈cps*(C) can have at most 2 singularities; those with no singularities are linear, those with a single singularity correspond to 𝒮_{p,α}, 0 < |α| < \frac{1}{4}π and ℛ_{p,α,β}, 0 < |β − \frac{1}{2}π| < \frac{1}{2}π, and those with two singularities correspond to the double Riemann and degenerate double Riemann nets 𝒟_{p,q,α,β,β} and ℱ_{p,q,σ,β,β} with 0 < |β − \frac{1}{2}π| < \frac{1}{2}π. All questions about the distribution of singularities of HP^{∗} -functions therefore have a direct but more restrictive counterpart for cps*-functions.

We next present a construction which shows that for any smoothly bounded Jordan domain D there are mappings in cps*(D) which have infinitely many singularities, but before going into the details, we make a few relevant comments. If D is a simply connected domain and if ℋ ∈ HP^{∗}(D) is known to have a singularity at some point p ∈ D then there can be no singularity at any point which is joined to p by a nonbounding characteristic of either of the fans associated with p in the case of a Riemann singularity or by any characteristic in the case of a spiral singularity. For this reason it is much easier for an HP-net to have many singularities if ∂D is highly contorted, since pockets formed by ∂D will allow characteristics emanating from p to hit the boundary before covering too much of D. For this reason it is easy to construct HP^{∗} -nets and cps*-mappings with infinitely many singularities (even with infinitely many spiral singularities), but such constructions will result in domains for which the boundary is not smooth. It is therefore of interest to see that the appearance of infinitely many singularities is possible even when the boundary is smooth; this follows immediately from the following

Theorem 5.2. There is a number M_{0} such that for all M > M_{0} there is an f ∈ cps^{∗}(H, 1, M), with Riemann singularities at 2^{k}z_{0} , −∞ < k < ∞, for some z_{0} ∈ H = {z : Im z > 0}.

Proof. For (small) ε > 0 let T = T (ε) denote the circular sector

\{1 + i tan^{−1}(\frac{1}{2}ε) + ρe^{iτ} : 0 < ρ ≤ 1, π + \frac{1}{2}ε ≤ τ ≤ 2π − \frac{1}{2}ε\}.

We begin by applying the characteristic coordinate constructions of Section 1.2 to obtain an HP^{∗} -net with an isolated singularity at p = 1 + i tan^{−1}(\frac{1}{2}ε) with C_{1} = \{|p|e^{it} : \frac{1}{2}ε ≤ t ≤ π + ε\} and C_{2} = [0, p] and fan angles ε and π − ε, respectively. For the resulting net the bounding characteristics of the 2-fan are the sides [p, 0] and [p, 2] of the sector T + 1 = {z + 1 : z ∈ T } and the bounding characteristics of the 1-fan are C_{1} and a convex arc C′ which joins p to a point b in the third quadrant (whose distance from e^{i(π+ε)} is O(ε)). Let F_{0} be the translate of C′ along the 2-arc formed by the segment [p, 2]; F_{0} is also a convex curve and joins 2 to some point c_{0} in the third quadrant. Note that the line segment 2-arc [b, c_{0}] is horizontal. The HP^{∗} -net so constructed is defined in the domain bounded by the 1-arc \{p + e^{iτ} : π + \frac{1}{2}ε ≤ τ ≤ 2π − \frac{1}{2}ε\} ∪ F_{0} and the 2-arc [c_{0}, b] ∪ bq ∪ [q, 0], where q = −e^{iε}. We now extend this net outward to larger and larger parts of H.

Assume inductively that we have a convex curve F_{k}, k ≥ 0, with the following properties:

(i) F_{k} starts at a_{k} = 6 · 2^{k} − 4 on the positive real axis and joins this point to a point c_{k} in the third quadrant.

(ii) The tangents to F_{k} at a_{k} and c_{k} have argument \frac{1}{2}(π − ε) and \frac{3}{2}π, respectively.

(iii) There is an HP^{∗} -net in a domain D_{k} which contains the region bounded by the part of F_{k} lying in the upper half-plane and a segment of the real line and in which the only singularities of the net are at the points 5 · 2^{m} − 4 + i2^{m} tan^{−1}(\frac{1}{2}ε), 0 ≤ m ≤ k.

(iv) F_{k} is a 1-characteristic of this net.

(v) Each point of F_{k} other than a_{k} is joined to 0 by a 2-characteristic of the net of (iii) which consists of a line segment [0, e^{iτ}], \frac{1}{2}ε < τ < π + ε, followed by a curve along which the argument of the tangent lies in the interval [τ − \frac{1}{2}ε, τ + \frac{1}{2}ε].

The arc F_{0} satisfies (i)–(v). A simple compactness argument shows that (v) implies that there is an ε_{0} , 0 < ε_{0} < 1/100, such that if ε < ε_{0} , then λ(F_{k}) < 2πa_{k}. (The number ε_{0} is an absolute constant which does not depend on k.) Henceforth we assume that ε < ε_{0} .

Let E_{k} be the circle of radius 2^{k} csc(\frac{1}{2}ε) centered at the point 7 · 2^{k} − 4 + i2^{k} cot(\frac{1}{2}ε) which joins the point a_{k} to 8 · 2^{k} − 4 in the lower half-plane. Note that the tangent to this arc at 8 · 2^{k} − 4 has argument \frac{1}{2}ε. We apply the usual characteristic quadrilateral HP-construction (Proposition 1.3) with F_{k} as the 1arc and

G_{k} = E_{k} ∪ [8 · 2^{k} − 4, 10 · 2^{k} − 4 + i2^{k+1} tan^{−1}(\frac{1}{2}ε)]

as the 2-arc. It is clear from the definition of ε_{0} that we have an a priori bound of 2π · 10 · 2^{k} = 5π · 2^{k+2} on the length of the translates of F_{k} along the 2-arc E_{k}, so that the concavity of E_{k} causes no singularity formation (that is, the mapping ζ of Proposition 1.3 is one-to-one in the whole rectangle I_{1} × I_{2} ) since the curvature of the curved part of this arc is 2^{−k} sin(\frac{1}{2}ε), and

2^{−k} sin(\frac{1}{2}ε) · 5π · 2^{k+2} < 10πε < 10πε_{0} < \frac{1}{10}π < 1.

Denote by F′_{k} the translate of F_{k} along G_{k}. We then put a singularity with 1- and 2-fan angles ε and π − ε, respectively, at 10 · 2^{k} − 4 + i2^{k+1} tan^{−1}(\frac{1}{2}ε), constructed in accordance with the procedures of Section 1.2, with 1-arc F′_{k} and 2-arc G_{k}. The 2-fan contains the entire sector T_{k+1} = 2^{k+1}T + 5 · 2^{k+1} − 4. Let F′′_{k} be the other bounding arc of the 1-fan, and let F_{k+1} be the translate of F′′_{k} along the right-hand segment of ∂T_{k+1} . One sees that F_{k+1} starts at 6 · 2^{k+1} − 4 and has the correct tangent angle there. For D_{k+1} we take D_{k} ∪ T_{k+1} together with the region covered by the translates of F_{k} along G_{k}, the 1-fan with bounding characteristics F′_{k} and F′′_{k} and the translates of F′′_{k} along the right-hand segment of ∂T_{k+1} . The 2-arcs lying in this new region and joining points of F_{k} and F_{k+1} consist of two line segments and two curved arcs which have total curvature (when reckoned as one moves away from the origin) ε and −ε, respectively. From this it follows that the terminal point of F_{k+1} indeed lies in the third quadrant and that its tangent there has argument \frac{3}{2}π. It is also clear that (v) holds for k + 1. The result of the entire inductive process is an HP^{∗} -net in all of H having singularities with dilatation (π − ε)/ε at the points 2^{k}(5 + i tan^{−1}(\frac{1}{2}ε)) − 4, k ≥ 0. Consideration of a convergent subsequence of {θ(2^{n}z) : n ≥ 0}, where θ is a corresponding HP-function, yields a HP^{∗} -net with singularities of this kind at the points 2^{k}(5 + i tan^{−1}(\frac{1}{2}ε)), −∞ < k < ∞, which, in turn gives the desired cps*-mapping.

The discussions of this and the preceding sections allow us to give some justification for identifying microscopic flaws in cryptocrystalline laminae with isolated singularities of the corresponding cps-mappings, or to be more precise, with isolated singularities of the HP-function θ corresponding to the inverse of the cpsmapping giving the deformation. Assume that the original uncrystallized lamina is represented by D ⊂ C and that the deformation produced by the crystallization is given by a homeomorphism f: D → C which is a cps-mapping on D\S. Here S is a closed set with components S_{k}, k ∈ I ⊂ N. The flaws will be the f(S_{k}). In the first place, because small flaws in close proximity to one another are apt to be perceived as a single larger flaw, it is reasonable to consider that a truly tiny flaw should be isolated as well as minute, that is, that for some T > 0, dist(f(S_{k}), f(S_{l})) ≥ T for all distinct k, l ∈ I and diam(f(S_{k})) < ε, k ∈ I, where ε > 0 is some suitably small number. A simple compactness argument shows that for given T, δ_{1}, δ_{2} > 0, there is an ε > 0 such that if θ ∈ HP(N(p, T)\F ) with F ⊂ N(p, ε), then there is a θ_{0} ∈ HP(N′(p, T)) such that

sup{|θ(z) − θ_{0}(z)| : δ_{1} < |z − p| < T } < δ_{2}.

In a neighborhood of f(S_{k}), θ_{f^{−1}} is therefore either close to an HP-function θ_{0} with an isolated singularity at some p ∈ f(S_{k}) or to one which is regular in that neighborhood, these two possibilities being mutually exclusive since, as our analysis shows, the θ_{0} with a true singularity at p are not approximable in any reasonable sense by functions in HP(N(p, T)). In the case that θ_{0} is regular we discard the flaw f(S_{k}) from consideration as such since its presence would produce no macroscopic effects. It would therefore appear that cps*-mappings provide a reasonable mathematical description of deformations induced by the solidification of planar liquid films which result in laminae known only to have microscopic flaws.

6. Further questions

In line with the geometric function theory focus outlined in the fourth paragraph of the introduction, the most immediate issues raised by the foregoing involve extremal questions about the possible distribution of the singularities of θ ∈ HP^{∗}(D) and f ∈cps*(D), and more specifically f ∈cps*(D, m_{1}, m_{2}). The cps-cases of this question have clear-cut significance in physical situations involving deformations with constant principal strains.

Since Corollary 4.1 says that for any Jordan domain D with smooth boundary the sum of the distances to the boundary of the singularities of any θ ∈ HP^{∗}(D) is bounded by 3λ(∂D)/π, it is most reasonable to ask about the smallest possible upper bounds for this sum, and for its analogues for cps*(D) and cps*(D, m_{1}, m_{2}). Although in general it is probably not possible to give an explicit answer to this question in terms of transparent geometric parameters of the domain D, it may well be in the case of disks; it is not clear, however, whether these problems have corresponding extremal functions. Somewhat more accessible, perhaps, is the determination of the greatest number of singularities that can appear in a given compact subset K of D; again the disk case, with K a concentric disk, is the most interesting instance. Theorem 5.2 tells us that given m_{1} , m_{2} with m_{1}/m_{2} > 1 sufficiently large, cps*(D, m_{1}, m_{2}) contains mappings with infinitely many singularities. One wonders if the condition that m_{1}/m_{2} be sufficiently large can be removed. Moreover, given the interpretation of cps-mappings in the context of cryptocrystalline laminae, it would be of considerable interest to determine if there always exist injective f ∈cps*(D, m_{1}, m_{2}) with infinitely many singularities, since noninjective f correspond to laminae which overlap themselves upon solidification. We believe that it should not be too difficult to resolve this issue. In a less quantitative direction we mention the possibility that if θ ∈ HP(D\S), where S is a “sufficiently small set” (linear measure 0 is perhaps small enough), then, in fact, θ can be extended to an HP-function in D\S′, where S′ ⊂ D consists entirely of isolated points of D.

The distribution of isolated singularities is only one of the numerous issues regarding HP-nets and cps-mappings which invite investigation. In addition to the distortion questions touched on in [G1], specific mapping problems offer another possibility. Here the analogy with conformal mappings becomes somewhat thin, since as will be shown in the sequel [G4] to this paper (see also [G3]), given any smoothly bounded Jordan domain D there is another such domain E such that for no f ∈cps(D) is f(D) = E. This raises the problem of finding general (but not necessarily exhaustive) intrinsic conditions on domains D and E which imply the existence of cps-homeomorphisms of one onto the other; particularly interesting is the question of when a domain has cps-self-homeomorphisms. An important tool for studying such mapping questions is the fact that if D and E are smoothly bounded domains and if f is a cps-homeomorphism of D onto E, then characteristics of the associated HP-function θ which meet ∂D do so at a well defined angle. This by no means trivial fact, to be proved in [G4], implies that the boundary values of θ are well defined (in an appropriate sense) and as a result allows one to relate such mapping questions to Cauchy problems. This, in turn, makes it possible to give simple descriptions of all cps-self-homeomorphisms of the half-plane and the exterior of a disk. All of these mapping questions have 238 Julian Gevirtz

corresponding cps*-versions and obvious interpretations in the context of cryptocrystalline laminae.

Constant principal strain mappings between 2-manifolds are treated in [ChG] and one can ask if it is possible to say something about the distribution of singularities in that context. In that paper a complete description of the infinite dimensional family of cps-self-homeomorphisms of the hyperbolic plane H^{2} is given, so that it would be interesting to find the H^{2} -analogue of Theorem 4.1. One might also ask if it is possible to derive some global results for higher dimensional mappings with constant principal strains, which are governed by a much more complex nonlinear hyperbolic system (see [G2]) or whether the results on HP-nets contained in this paper can be generalized to families of nets associated with other 2 × 2 genuinely nonlinear hyperbolic systems.

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Received 7 May 1998