Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 239–270

GEOMETRIC STRUCTURE OF TUBES AND BANDS OF ZERO MEAN CURVATURE IN MINKOWSKI SPACE

V. A. Klyachin and V. M. Miklyukov

Volgograd State University, Department of Mathematics 2 Prodolnaya 30, RU-400062 Volgograd, Russia; klchnv@mail.ru; miklyuk@mail.ru

Abstract. Spacelike and timelike tubes and bands of zero mean curvature in Minkowski space are investigated in a neighborhood of finite or infinite singularities. We also study the correlation between the branching of surfaces and their exterior amounts, and questions of the smooth pasting of spacelike and timelike tubes and bands. We give an asymptotic representation of the surfaces in the neighborhood of the singular point.

1. Introduction

We investigate the structure of tubes and bands of zero mean curvature in Minkowski space in a neighborhood of their singularity. An important property of the case we study is the existence of isolated singularities of cone type [20], [11] and [18]. This property is specific for surfaces in Lorentz spaces.

Geometric and topological aspects of the structure of Lorentzian manifolds having singularities were investigated in [15] and [2].

We also consider another problem. That is, we study manifolds with singularities embedded in Minkowski space. Firstly, we are interested in questions connected with the exterior structure of manifolds in a neighborhood of their singular points. We also consider some questions connected with processes of transition from spacelike to timelike manifolds at their common singular point.

We give some results in the following directions: to describe the exterior structure of spacelike bands with infinite number of branches at the infinity of R_{1}^{n+1} ; to obtain an asymptotic decomposition of zero mean curvature tubes and bands in the neighborhood of singular points; to investigate possibilities of the smooth pasting of spacelike tubes and bands with timelike ones at the singular point.

It is possible that Shiffman [33], Nitsche [31], and Osserman and Shiffer [32] were the first to investigate tubes of zero mean curvature. The minimal surfaces of a tubular type of arbitrary codimension in R^{n+1} were defined in [22] and minimal bands were introduced in [24]. The idea of investigating bands was borrowed from the theory of relative strings (for example, see [3] and [7]), where tubes and bands of zero mean curvature in Minkowski space R_{1}^{n+1} (but with timelike and not Riemannian structure, induced by the scalar product of R_{1}^{n+1} ) are important objects of research.

2000 Mathematics Subject Classification: Primary 53C50, Secondary 53C80.

From the geometric viewpoint, relativistic strings and membranes are surfaces of zero mean curvature in Minkowski space-time. The surfaces of tubular type correspond to some closed strings there. The open strings are interpreted as bands of a special form. An approach to the string theory from the viewpoint of their geometric structure is promising, since even the simplest extrinsic properties of a surface in the framework of that or other models can be translated into the language of physical phenomena. Thus, the estimate of the extension of a tube or a band along the time-axis corresponds to the estimate of the lifetime of the string; the projection e_{0}^{T} (m) of the time vector e_{0} onto the tangent plane T (m) of the surface ℳ at the point m ∈ M corresponds to the local time on the string; branch points of surfaces correspond to the beginning of change in the type of a physical process, decay of particles, and so on [27].

The ideas of string theory lie on the basis of a Nilsen conjecture [30], which states that metrics of zero mean curvature in Minkowski space are only physically significant among all metrics which are solutions of the Einstein equation. The fact that these surfaces have isolated singular points in R_{1}^{n+1} ensures possibilities for modeling some special aspects of the ‘big bang’ [15] by the tubes and bands of zero mean curvature surfaces. By analogy with the ‘big bang’, the problem of pasting is an attempt to answer the question: what could exist before the ‘big bang’ of a universe?

Now in spite of the many papers devoted to relativistic strings and their generalizations, there is no mathematical theory of strings. We regard the construction of this theory as a superproblem and disregard important questions of strings quantification. We restrict ourselves to a more narrow set of questions: describing the geometric structure of the string, namely, investigating the structure of spacelike and timelike tubes and bands in Minkowski space.

Among the papers devoted to the structure of zero mean curvature tubes and bands in a neighborhood of a singular point, we distinguish the pioneering paper [20], where it is shown that the set of the tangent rays to any maximal surface in a neighborhood of an essentially singular point coincides with upper or lower sheets C^{+} or C^{−} of the light cone (also, see [11]).

In our papers [18] and [21] this result was sharpened. We gave quantitative characteristics of this property in terms of interior and exterior girth functions of tubes.

We do not know any results relevant to asymptotic decompositions of zero mean curvature surfaces in a neighborhood of a singularity or to pasting problems.

For the structure of zero mean curvature surfaces in Minkowski space, see also [8], [13], [4], [5], [10], [12], and [25].

Acknowledgments. The work is supported by RFFI, project 97-01-00414 and INTAS, project 10170.

The authors wish to thank Dr. Tyler, J. Jarvis and Dr. E. A. Pelich who corrected the initial English text. We also thank Dr. V. G. Tkachev for useful remarks during the preparation of the article.

The second author wishes to thank the Department of Mathematics of Brigham Young University for hospitality while the English version of this manuscript was written.

2. Main results

Let R_{1}^{n+1} be a Minkowski space, that is, an (n + 1)-dimensional real pseudo-Euclidean space with a metric of signature (1, n). Let x = (x_{1}, x_{2}, …, x_{n}) ∈ R^{n}, t ∈ R^{1} and χ = (t, x) ∈ R_{1}^{n+1} . For an arbitrary pair of vectors χ′ = (t′, x′) and χ′′ = (t′′, x′′) of R_{1}^{n+1}, we denote their scalar product by

(2.1)  ⟨χ′, χ′′⟩ = −t′t′′ + ∑_{i=1}^{n} x′_{i}x′′_{i}.

The scalar square of a vector χ ∈ R_{1}^{n+1} is

|χ|^{2} = −t^{2} + ∑_{i=1}^{n} x_{i}^{2}.

A nonzero vector χ ∈ R_{1}^{n+1} is called spacelike, lightlike, or timelike if |χ|^{2} > 0, |χ|^{2} = 0, or |χ|^{2} < 0, respectively. The totality C = C(χ_{0}) of the lightlike vectors with origin at a point χ_{0} ∈ R_{1}^{n+1} forms the light cone. We shall denote upper and lower sheets of the light cone by C^{+} = C^{+}(χ_{0}) and C^{−} = C^{−}(χ_{0}).

Let M be a two-dimensional connected, orientable noncompact manifold of C^{2} with a piecewise smooth boundary ∂M (possibly empty). Consider the surface ℳ = (M, u) given by a C^{2} -immersion χ = u(m): M → R_{1}^{n+1} .

The surface ℳ = (M, u) is said to be spacelike if its tangent vectors are spacelike. If the surface ℳ is spacelike, then the scalar product (2.1) induces a Riemannian metric on ℳ , and the standard connection \overline{∇} in R_{1}^{n+1} induces a Riemannian connection ∇ on ℳ . In addition, the Riemannian metric on M and the connection ∇ are coordinated [6, Addition A]. By Δ we will denote the Laplacian in this metric.

The surface ℳ = (M, u) is said to be timelike if for each point m ∈ M the tangent plane T_{u(m)} contains both spacelike and timelike vectors.

Let {e_{i}}_{i=0}^{n} be an orthonormal basis in R_{1}^{n+1} for which

⟨e_{i}, e_{j}⟩ = 0 for i ≠ j, |e_{0}|^{2} = −1 and |e_{i}|^{2} = 1 for i = 1, 2, …, n.

Therefore, χ = te_{0} + ∑_{i=1}^{n} x_{i}e_{i}.

We denote a hyperplane of constant time by

Π(τ) = {χ ∈ R_{1}^{n+1} : ⟨χ + τe_{0}, e_{0}⟩ = 0}.

A surface ℳ = (M, u) in R_{1}^{n+1} is called a band with projection (α, β), −∞ ≤ α < β ≤ +∞, with the time axis 0t if it satisfies the following properties:

(a) for any τ_{1}, τ_{2} ∈ (α, β) the set

M(τ_{1}, τ_{2}) = {m ∈ M : τ_{1} < t(m) < τ_{2}}, t(m) = −⟨u(m), e_{0}⟩,

is precompact; (b) for any τ ∈ (α, β) the intersection Σ(τ) = u(M) ∩ Π(τ) is not empty;

(c) there exists τ ∈ (α, β) such that at least one of the connected components of u^{−1}(Σ(τ)) contains points of ∂M;

(d) any vector ν of the unit normal to u(∂M) on u(M) satisfies ⟨ν, e_{0}⟩ = 0;

(e) for any point m ∈ ∂M at which the boundary ∂M does not have any tangent plane, the contingency contg_{u(m)}u(M) does not contain lightlike rays.

Sometimes it is necessary to use the following property: (c′) each connected component of the set u^{−1}(Σ(τ)) contains points of ∂M. This condition is stronger than (c). A surface ℳ = (M, u) in R_{1}^{n+1} is called

a strict band if it satisfies (a), (b), (c′), (d) and (e).

The (finite or infinite) quantity β − α is said to be the time existence (length) of the band.

The surface ℳ = (M, u) in R_{1}^{n+1} is said to be a surface of tubular type with a projection (α, β) if M is a manifold without a boundary and (M, u) has properties (a) and (b).

A tube or a band is called the tube or band in large if its projection is (−∞, +∞).

Some examples of tubes and bands of zero mean curvature R_{1}^{n+1} can be found in [18], [19] and [27].

Let ℳ = (M, u) be a surface and C(χ_{0}) be a light cone. If for some m ∈ M

l^{2}(m, χ_{0}) = −(u_{0}(m) − t_{0})^{2} + ∑_{i=1}^{n}(u_{i}(m) − x_{0 i})^{2} < 0,

then the point u(m) lies inside C(χ_{0}).

If for some m ∈ M the magnitude l^{2}(m, χ_{0}) > 0, then u(m) lies outside C(χ_{0}).

Suppose that for some χ_{0}

(2.2)  lim sup_{t(m)→α} l^{2}(m, χ_{0}) ≤ 0,

and that the set M^{+} = {m ∈ M : l^{2}(m, χ_{0}) > 0} is not empty. For arbitrary p, q > 0, we consider a counting function N_{u}(t) = N_{u}(t; p, q) of the (p, q)- connected components of M^{+} which are defined by (3.34). Roughly speaking, the number N_{u}(t) denotes the number of the (p, q)-components of M^{+} ∩ Σ(t).

Next we denote a flow of time through Σ(t) by

μ(t) = ∫_{Σ(t)} |∇t(m)|.

If ℳ is a tube or band of zero mean curvature, then μ is independent of t [27].

2.3. Theorem. Let ℳ be a two-dimensional tube or band of zero mean curvature with the projection (α, ∞) and the condition (2.2).

If the set M^{+} is not empty, then for any p, q > 0 and arbitrary τ > α, it is true that

N_{u}(τ) exp \{\frac{1}{μN_{u}(τ)} ∫_{α}^{τ} N_{u}^{2}(t) dt\} ≤ 4μ max_{m∈Σ(τ+1)} l^{4}(m, χ_{0}).

Fix an arbitrary vector e ∈ R_{1}^{n+1} and consider the function h(m) = ⟨u(m), e⟩. Suppose that

(2.4)  lim sup_{t(m)→α} h(m) ≤ 0,

and that the set M^{+} = {m ∈ M : h(m) > 0} is not empty.

The following theorem strengthens a corresponding theorem in [27], which treats the case that M^{+} has a finite number of connected components.

2.5. Theorem. Let ℳ be a band or tube of zero mean curvature with a projection (α, ∞) and with (2.4).

For any p, q > 0 and an arbitrary τ > α, it is true that

(2.6)  N_{u}(τ) exp \{\frac{1}{μN_{u}(τ)} ∫_{α}^{τ} N_{u}^{2}(t) dt\} ≤ 4μ max_{m∈Σ(τ+1)} h^{2}(m).

Both Theorem 2.3 and 2.5 are geometric corollaries of the more general Theorem 3.39 for arbitrary solutions of (3.18) on ℳ .

Let G ⊂ R^{2} be a domain in the plane and let (0, 0) ∈ G. We consider a solution f ∈ C^{2} of the equation

(2.7)  \frac{∂}{∂x}(\frac{f_{x}}{\sqrt{1 − f_{x}^{2} − f_{y}^{2}}} ) + \frac{∂}{∂y} (\frac{f_{y}}{\sqrt{1 − f_{x}^{2} − f_{y}^{2}}} ) = 0.

This equation describes the spacelike zero mean curvature surfaces in Minkowski space-time.

We assume that f is defined on G and has an isolated singularity at the origin (0, 0). Ecker [11] has shown that

|f(x, y) − f(0, 0)| ∼ \sqrt{x^{2} + y^{2}} as (x, y) → (0, 0).

In [18] and [27], we sharpened this asymptotic. That is, we proved that

(2.8)  ν̄ = 6 lim sup_{x,y→0} \frac{\sqrt{x^{2} + y^{2}} − |f(x, y) − f(0, 0)|}{(x^{2} + y^{2})^{3/2}} < ∞,

and, moreover, ν̄ ≥ μ^{−2} .

Let

κ^{∗}(x, y) = κ^{∗}(ϱe^{iψ}) = \frac{K(x, y)}{sinh^{4} α(x, y)} = \frac{f_{xy}^{2} − f_{xx}f_{yy}}{(f_{x}^{2} + f_{y}^{2})^{2}}

be the curvature expression in the coordinates x + iy = ϱe^{iψ}.

2.9. Theorem. Let f(x, y) be a solution of (2.7) with a singularity at (0, 0). Then,

(1) there exist 2π -periodic real analytic functions c_{k}(ψ) defined on [0, 2π] such that the following decomposition holds:

(2.10)  f(ϱe^{iψ}) = ϱ + ∑_{k=1}^{∞} c_{k}(ψ)ϱ^{2k+1};

(2) there exists a limit

(2.11)  lim_{x,y→0} κ^{∗}(x, y) = lim_{ϱ→0} κ^{∗}(ϱe^{iψ}) = −6c_{1}(ψ) = κ^{∗}(ψ);

(3) the following equalities are true:

(2.12)  6 lim_{ϱ→0} \frac{ϱ − f(ϱe^{iψ})}{ϱ^{3}} = κ^{∗}(ψ)

and

(2.13)  ∫_{0}^{2π} \frac{dψ}{\sqrt{κ^{∗}(ψ)}} = μ.

Further, we suppose that f(x, y) is a solution of timelike zero mean curvature surfaces equation in the Minkowski space-time

(2.14)  \frac{∂}{∂x}(\frac{f_{x}}{\sqrt{f_{x}^{2} + f_{y}^{2} − 1}}) + \frac{∂}{∂y} (\frac{f_{y}}{\sqrt{f_{x}^{2} + f_{y}^{2} − 1}}) = 0.

We assume f is defined on a domain G ⊂ R^{2} and has an isolated singularity at (0, 0).

The following statement is similar to a known theorem for spacelike zero mean curvature surfaces [11], which says that the totality of the tangent rays in the singular point forms an upper or lower sheet of the light cone.

2.15. Theorem. Let ℳ be a two-dimensional timelike tube of zero mean curvature with a singularity at {0} ∈ R_{1}^{3} . The tangent rays to ℳ at this point are lightlike.

Let f_{1}(x, y) and f_{2}(x, y) be solutions of (2.7) and (2.14), respectively, having isolated singularities at (0, 0) such that

f_{1}(0, 0) = f_{2}(0, 0) = 0,  f_{1}(x, y) < 0,  f_{2}(x, y) > 0.

We will say that solutions f_{1} and f_{2} are C^{k} -pasted if

δ(ϱe^{iψ}) ∈ C^{k}, where δ(ϱe^{iψ}) = f_{1}(ϱe^{iψ}) + f_{2}(ϱe^{iψ}).

The following statement asserts the possibility of the smooth gluing of solutions.

2.16. Theorem. For an arbitrary solution f_{1}(x, y) of (2.7) with a singularity at (0, 0), there exists a solution f_{2}(x, y) of (2.14) with a singularity at (0, 0) such that f_{1}(x, y) and f_{2}(x, y) are C^{2} -pasted.

3. A structure at infinity

Let ℳ = (M, u), dim M = 2, be a spacelike tube or band of zero mean curvature in R_{1}^{n+1} with projection (α, β) onto the time axis 0t. Below we will use the notation and terminology of [27, Section 3]. That is to say, we will need a concept of the ends ξ_{M}(α), ξ_{M}(β) of the surface ℳ which are determined by analogy with the prime ends of a planar domain (for example, see [34]).

Let f(m), f(m) ≢ 0, be an arbitrary function of C^{0}(\overline{M}) ∩ C^{2}(M) such that

(3.17)  f|_{∂M} = 0.

We suppose

(3.18)  fΔf ≥ 0 everywhere on M.

The differential inequality (3.18) is not traditional. We give some simple properties of functions f satisfying (3.17) and (3.18).

At first, solutions of (3.17) and (3.18) do not have a strict maximum in the domain.

3.19. Lemma. Let f be a solution of (3.18) satisfying (3.17). Any connected component 𝒪 of the set {m ∈ M : f(m) > 0} does not have a compact closure \overline{𝒪} .

Proof. We suppose that the closure \overline{𝒪} is compact. By Gauss’ formula, we have

∫_{∂𝒪} f⟨∇f, ν⟩ = ∫_{𝒪} |∇f|^{2} + ∫_{𝒪} fΔf,

where ν is a unit outward normal to ∂𝒪.

Since f|_{∂𝒪} = 0, the contour integral vanishes. Therefore, from (3.18) it follows that

∫_{𝒪} |∇f|^{2} = 0.

Hence f ≡ const on 𝒪 contradicts the definition of 𝒪.

These arguments are strict only if the boundary ∂𝒪 is rectifiable. In the general case, the function f is extended by zero outside \overline{𝒪} . Further, the function obtained can be approximated by C^{2} -smooth, compactly-supported functions on M.

We denote

osc{f, Σ(t)} = sup_{x,y∈Σ(t)} |f(x) − f(y)|.

3.20. Lemma. If

(3.21)  lim inf_{t→α} osc {f, Σ(t)} = 0,

then

(3.22)  lim_{m→ξ_{M} (α)} f(m) = 0,

and for an arbitrary connected component 𝒪, there exists t_{0} ∈ (α, β) such that

(3.23)  𝒪 ∩ Σ(t) ≠ ∅  for all t > t_{0}.

Proof. The proof follows from the weak maximum-minimum principle for solutions of differential inequalities (3.18). The maximum principle is proved by Lemma 3.19. In order to prove the minimum principle, it is sufficient to note that both f and −f satisfy (3.18).

We fix an arbitrary connected component 𝒪 and suppose that the relation (3.21) holds. We denote by τ(𝒪) the smallest among values t_{0} ≥ α for which (3.23) is true. For each fixed t > τ(𝒪), let γ_{1}(t), γ_{2}(t), … be all connected components of 𝒪 ∩ Σ(t) having properties: \overline{γ_{i}(t)} ∩ ∂M ≠ ∅, i = 1, 2, ….

We denote

μ(t, 𝒪) = sup_{i} ∫_{γ_{i}(t)} |∇t(m)|.

For an arbitrary t > τ(𝒪), we put

J(t, 𝒪) = ∫_{𝒪∩M(α,t)} |∇f|^{2},

where M(t_{1}, t_{2}) = {m ∈ M : t_{1} < t(m) < t_{2}}.

The following lemma is basic in the present paragraph.

3.24. Lemma. If a function f(m) satisfies (3.17), (3.18) and (3.21), then for almost every t > τ(𝒪) we have

(3.25)  J(t, 𝒪) ≤ \frac{μ(t, 𝒪)}{2π} J′(t, 𝒪).

Proof. Using the Stokes formula and (3.17), we can write

∫_{𝒪∩Σ(t)} f⟨∇f, ν⟩ = ∫_{∂(𝒪∩M(α,t))} f⟨∇f, ν⟩ = J(t, 𝒪) + ∫_{𝒪∩M(α,t)} f Δf,

where ν is an inward normal to the boundary of 𝒪 ∩ M(α, t) on M. From (3.18) we obtain

(3.26)  J(t, 𝒪) ≤ ∫_{𝒪∩Σ(t)} f ⟨∇f, ν⟩.

As

ν = \frac{∇t}{|∇t|}(m) for all m ∈ Σ(t),

the condition (3.17) and Cauchy’s inequality give

(3.27) \begin{aligned}∫_{𝒪∩Σ(t)} f ⟨∇f, ν⟩ & = ∫_{𝒪∩Σ(t)} f ⟨∇f, ∇t⟩ \frac{1}{|∇t|} = ∑_{i} ∫_{γ_{i}(t)} f ⟨∇f, ∇t⟩\frac{1}{|∇t|} \\ & ≤ ∑_{i}(∫_{γ_{i}(t)} f^{2} |∇t|)^{1/2}(∫_{γ_{i}(t)}⟨∇f, \frac{∇t}{|∇t|}⟩^{2} \frac{1}{|∇t|})^{1/2}. \end{aligned}

The following arguments are close to arguments from [27, Lemma 5.1]. Let γ = γ_{i}(t) be an arbitrary connected component of 𝒪∩Σ(t) and m(s): [0, length (γ)] → γ be its natural parameterization. We put

v(s) = ∫_{0}^{s}|∇t(m(s))| ds,  ṽ = v(length (γ)).

Since the arc γ = γ_{i}(t) is open and γ_{i} ∩ ∂M ≠ ∅, we get from (3.17)

f|_{s=0} = f|_{s=length (γ)} = 0.

By Wirtinger’s inequality,

∫_{γ} f^{2}|∇t| = ∫ ṽ ; 0 f^{2} dv(s) ≤ (\frac{ṽ}{π} )^{2} ∫ ṽ ; 0(\frac{df}{dv} )^{2} dv = (\frac{ṽ}{π} )^{2} ∫_{γ}(\frac{df}{ds})^{2} \frac{ds}{|∇t|}.

Now we have

ṽ = ∫_{γ} |∇t| ≤ sup_{i} ∫_{γ_{i}} |∇t| ≤ μ(t, 𝒪),

and, therefore,

∫_{γ} f^{2}|∇t| ≤ \frac{μ^{2}(t, 𝒪)}{π^{2}} ∫_{γ}(\frac{df}{ds})^{2} \frac{1}{|∇t(m)|}.

This relation is true for any open arc γ = γ_{i}(t) such that γ_{i} ∩ ∂M ≠ ∅. From (3.26) and (3.27), we find that

J(t, 𝒪) ≤ \frac{μ(t, 𝒪)}{π} ∑_{i}(∫_{γ_{i}(t)}(\frac{df}{ds})^{2} \frac{1}{|∇f|})^{1/2}(∫_{γ_{i}(t)}⟨∇f, \frac{∇t}{|∇t|}⟩^{2} \frac{1}{|∇t|})^{1/2}

≤ \frac{μ(t, 𝒪)}{2π} ∑_{i} ∫_{γ_{i}(t)}((\frac{df}{ds})^{2} + ⟨∇f, \frac{∇t}{|∇t|}⟩^{2})\frac{1}{|∇t|}

≤ \frac{μ(t, 𝒪)}{2π} ∫_{𝒪∩Σ(t)} ((\frac{df}{ds})^{2} + ⟨∇f, \frac{∇t}{|∇t|}⟩^{2})\frac{1}{|∇t|}.

At each point m ∈ 𝒪 ∩ Σ(t), we have

(\frac{df}{ds})^{2} + ⟨∇f, \frac{∇t}{|∇t|}⟩^{2} = |∇f|^{2}.

Therefore, from the previous inequality we obtain

(3.28)  J(t, 𝒪) ≤ \frac{μ(t, 𝒪)}{2π} ∫_{𝒪∩Σ(t)} |∇f|^{2} \frac{1}{|∇t|}.

Now we use the following co-area formula for integration over level sets of t = t(m):

J(t, 𝒪) = ∫_{τ(𝒪)}^{t} dτ ∫_{𝒪∩Σ(τ)} |∇f|^{2} \frac{1}{|∇t|}.

Hence, for almost every t ∈ (τ(𝒪), ∞) we have

J′(t, 𝒪) = ∫_{𝒪∩Σ(t)} |∇f|^{2} \frac{1}{|∇t|}.

Combining this relation with (3.28), we get (3.25).

3.29. Definition. Let p, q > 0 be an arbitrary pair of numbers. We will say that a connected component 𝒪 of f has the type (p, q) (or it is a (p, q)-component) if for any t > τ(𝒪) + p,

max_{m∈Σ(t)∩𝒪} |f(m)| ≥ q.

3.30. Lemma. If a domain 𝒪 has the type (p, q), then for any t > τ(𝒪)+p′ with p′ > p, it is true that

(3.31)  \frac{(p′ − p)q^{2}}{μ} exp\{∫_{τ(𝒪)+p′}^{t} \frac{ds}{μ(s)}\} ≤ J(t, 𝒪),

where

μ = ∫_{Σ(t)} |∇t(m)|

does not depend on t and μ(s) = μ(s, 𝒪).

Proof. From (3.25) for any t > τ(𝒪) + p′, we can write

∫_{τ+p′}^{t} \frac{ds}{μ(s)} ≤ log \frac{J(t, 𝒪)}{J(τ + p′, 𝒪)}

with τ = τ(𝒪) and μ(s) = μ(s, 𝒪). Therefore,

(3.32)  J(τ + p′, 𝒪) exp\{∫_{τ+p′}^{t} \frac{ds}{μ(s)}\} ≤ J(t, 𝒪).

As the connected component 𝒪 has the type (p, q), then for any t > τ(𝒪)+p, we have

\begin{aligned}q^{2} & ≤ max_{m∈Σ(t)∩𝒪} |f(m)|^{2} ≤ (∫_{Σ(t)∩𝒪} |∇f(m)|)^{2} \\ & ≤ (∫_{Σ(t)∩𝒪} |∇f(m)|^{2} \frac{1}{|∇t(m)|})(∫_{Σ(t)∩𝒪} |∇t(m)|). \end{aligned}

By the property (d) of the band, we have everywhere on ∂M that

⟨∇t, ν⟩ = −⟨e_{0}, ν⟩ = 0.

Fix numbers t_{1} < t_{2} so that α < t_{1} < t_{2} < ∞. We have

\begin{aligned}∫_{∂M(t_{1},t_{2})}⟨∇t, ν⟩ & = ∫_{Σ(t_{2})}⟨∇t, ν⟩ − ∫_{Σ(t_{1})}⟨∇t, ν⟩ \\ & = ∫_{Σ(t_{2})} |∇t| − ∫_{Σ(t_{1})} |∇t| = ∫_{M(t_{1},t_{2})} Δt = 0. \end{aligned}

250 V. A. Klyachin and V. M. Miklyukov Consequently, the integral

μ = ∫_{Σ(t)} |∇t|

is independent of t. Now we obtain

\frac{(p′ − p)q^{2}}{μ} ≤ q^{2} ∫_{τ+p}^{τ+p′} \frac{ds}{μ(s)} = q^{2} ∫_{τ+p}^{τ+p′} ds / ∫_{Σ(s)} |∇t(m)|

≤ q^{2} ∫_{τ+p}^{τ+p′} ds / ∫_{Σ(s)∩𝒪} |∇t(m)|

≤ ∫_{τ+p}^{τ+p′} ds ∫_{Σ(s)∩𝒪} |∇f(m)|^{2} \frac{1}{|∇t(m)|} ≤ J(τ + p′, 𝒪).

Taking into account (3.32), we arrive at (3.31).

Now, we let {m ∈ M : f(m) > 0} have either a finite or an infinite number of connected components 𝒪_{1}, 𝒪_{2}, … of the type (p, q).

Clearly, for any given finite number of connected components 𝒪_{1}, 𝒪_{2}, …, 𝒪_{N} of {m ∈ M : f(m) > 0}, there are p, q > 0 such that all 𝒪_{i} are (p, q)-components.

We set

τ_{i} = τ(𝒪_{i}), μ_{i}(t) = μ(t, 𝒪_{i}), i = 1, 2, …,

and define functions μ_{i}^{∗}(t): (α, ∞) → R, i = 1, 2, …, in the following way:

μ_{i}^{∗}(t) = μ_{i}(t) for t > τ_{i} + p + 1 and μ_{i}^{∗}(t) = ∞ for t ∈ (α, τ_{i} + p + 1].

In Lemma 3.30, we choose p′ = p + 1 and fix t > α. Since (3.31) for any i = 1, 2, … such that τ_{i} + p + 1 < t, we can write

\frac{q^{2}}{μ} exp\{∫_{τ_{i}+p+1}^{t} \frac{ds}{μ_{i}(s)}\} ≤ ∫_{𝒪_{i}∩M(α,t)} |∇f|^{2},

and

(3.33)  \frac{q^{2}}{μ} ∑_{i=1}^{N} exp\{∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(s)}\} ≤ ∑_{i=1}^{N} ∫_{𝒪_{i}∩M(α,t)} |∇f|^{2} ≤ ∫_{M(α,t)} |∇f|^{2}.

3.34. Definition. The counting function N_{f}(t; p, q) is equal to the number of all (p, q)-domains 𝒪_{i} for which τ(𝒪_{i}) + p + 1 < t, and it vanishes for t ≤ inf_{i} τ(𝒪_{i}) + p + 1.

Geometric structure of tubes and bands of zero mean curvature 251 We use an inequality between arithmetic and geometric means. We have

exp \{\frac{1}{N_{f}(t)} ∑_{i=1}^{N_{f} (t)} ∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(s)}\} = ( ∏_{i=1}^{N_{f} (t)} exp\{∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(s)}\})^{1/N_{f} (t)}

≤ \frac{1}{N_{f}(t)} ∑_{i=1}^{N_{f} (t)} exp\{∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(t)}\},

where N_{f}(t) = N_{f}(t; p, q) is the counting function.

Hence from (3.33) for any t > α, we get

(3.35)  \frac{q^{2}}{μ} N_{f}(t) exp\{\frac{1}{N_{f}(t)} ∑_{i=1}^{N_{f} (t)} ∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(s)}\} ≤ ∫_{M(α,t)} |∇f|^{2}.

Further, we note that

∑_{i=1}^{N_{f} (t)} ∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(s)} = ∫_{α}^{t} ∑_{i=1}^{N_{f} (t)} \frac{ds}{μ_{i}^{∗}(s)} = ∫_{α}^{t} ∑_{i=1}^{N_{f} (s)} \frac{ds}{μ_{i}^{∗}(s)}

≥ ∫_{α}^{t} N_{f}(s)( ∏_{i=1}^{N_{f} (s)} \frac{1}{μ_{i}^{∗}(s)})^{1/N_{f} (s)} ds

= ∫_{α}^{t} N_{f}(s) ds / ( ∏_{i=1}^{N_{f} (s)} μ_{i}^{∗}(s))^{1/N_{f} (s)}

≥ ∫_{α}^{t} N_{f}^{2}(s) ds / ∑_{i=1}^{N_{f} (s)} μ_{i}^{∗}(s) ≥ ∫_{α}^{t} N_{f}^{2}(s) ds / ∑_{i=1}^{N_{f} (s)} ∫_{Σ(s)∩𝒪_{i}} |∇t|

≥ ∫_{α}^{t} N_{f}^{2}(s) ds / ∫_{Σ(s)} |∇t| ≥ \frac{1}{μ} ∫_{α}^{t} N_{f}^{2}(s) ds,

and, consequently,

(3.36)  ∑_{i=1}^{N_{f} (t)} ∫_{α}^{t} \frac{ds}{μ_{i}^{∗}(s)} ≥ \frac{1}{μ} ∫_{α}^{t} N_{f}^{2}(s) ds.

Combining (3.35) and (3.36), we obtain the following estimate

(3.37)  \frac{q^{2}}{μ} N_{f}(t) exp \{\frac{1}{μN_{f}(t)} ∫_{α}^{t} N_{f}^{2}(s) ds\} ≤ ∫_{M(α,t)} |∇f|^{2},

where N_{f}(t) = N_{f}(t; p, q).

Therefore, we obtain the following assertion which is important for applications.

3.38. Lemma. Let f(m) be a C^{2} -solution of (3.18) on ℳ satisfying (3.17) and (3.21). Let p, q > 0 and let 𝒪_{1}, 𝒪_{2}, … be (p, q)-components of {m ∈ M : f(m) > 0}. Then for any t > α, the counting function N_{f}(t) = N_{f}(t; p, q) < ∞ is nondecreasing, and (3.37) is true.

For the proof it is sufficient to show that the counting function is locally finite. By Lemmas 3.19 and 3.20 for each connected component 𝒪_{i}, and for any t > τ(𝒪_{i}), it is true that 𝒪_{i} ∩ Σ(t) ≠ ∅. Thus the function N_{f}(t) is nondecreasing as t → ∞. The relation (3.37) implies locally the boundedness of N_{f}(t).

3.39. Theorem. Let f(m) be a C^{2} -solution of (3.18) on ℳ satisfying (3.17) and (3.21). Let p, q > 0 and let 𝒪_{1}, 𝒪_{2}, … be (p, q)-components of {m ∈ M : f(m) > 0}. Then for any τ > α, it is true that

(3.40)  N_{f}(τ) exp \{\frac{1}{μN_{f}(τ)} ∫_{α}^{τ} N_{f}^{2}(t) dt\} ≤ \frac{μ^{2}}{q^{2}} max_{m∈Σ(τ+1)} f^{2}(m),

where N_{f}(t) = N_{f}(t; p, q).

Proof. Let ϕ(m) = ξ∘t(m), and let ξ(t) = 1 for α < t < τ and ξ(t) = τ +1−t for τ ≤ t ≤ τ + 1. By (3.17), (3.18) and (3.21), we can write

∫_{M(α,τ)} |∇f|^{2} ≤ ∫_{M(α,τ+1)} ϕ^{2}|∇f|^{2}

= ∫_{∂M(α,τ+1)}fϕ^{2}⟨∇f, ν⟩ − 2 ∫_{M(α,τ+1)}fϕ⟨∇f, ∇ϕ⟩ − ∫_{M(α,τ+1)}ϕ^{2}fΔf

≤ −2 ∫_{M(α,τ+1)} fϕ⟨∇f, ∇ϕ⟩

≤ 2(∫_{M(α,τ+1)} f^{2}|∇ϕ|^{2})^{1/2}(∫_{M(α,τ+1)} ϕ^{2}|∇f|^{2})^{1/2}.

Thus, we find

∫_{M(α,τ+1)} ϕ^{2}|∇f|^{2} ≤ 4 ∫_{M(α,τ+1)} f^{2}|∇ϕ|^{2}

and

∫_{M(α,τ)} |∇f|^{2} ≤ 4 max_{m∈Σ(τ+1)} f^{2}(m) ∫_{M(τ,τ+1)} |∇t|^{2}

= 4 max_{m∈Σ(τ+1)} f^{2}(m) ∫_{τ}^{τ+1} dt ∫_{Σ(t)} |∇t|

≤ 4μ max_{m∈Σ(τ+1)} f^{2}(m).

Using (3.37), we arrive at (3.40).

Proof of Theorem 2.3. At first we recall that the function

l^{2}(m) = |u(m) − χ_{0}|^{2}

satisfies the inequality Δl > 0.

Let k ∈ R_{1}^{n} be a fixed vector, and let ℳ ⊂ R_{1}^{n+1} be a two-dimensional surface. For an arbitrary point m ∈ M, we denote by T = T (m) the tangent space to ℳ at this point and the projection of k onto T (m) by k^{T} = k^{T} (m). It is not difficult to see that

Δl^{2}(m) = −2|e_{0}^{T} |^{2} + 2 ∑_{i=1}^{n} |e_{i}^{T} |^{2} = 4.

If {m ∈ M : f(m) > 0} = M^{+} is not empty, then l^{2}Δl^{2} ≥ 0 on M^{+} , and (2.2) implies (3.40) for any p, q > 0.

Proof of Theorem 2.5. Here h(m) satisfies Δh = 0 on M and hΔh ≥ 0 on M^{+} . By Theorem 3.39, property (2.4) implies (3.21) and it follows that (3.40) and (2.6) also hold.

4. Neighborhood of isolated singularity

Let ℳ = (M, u) be a tube with a projection (α, β) defined by an immersion u: M → R_{1}^{3} . We say that the surface ℳ has a singularity at χ_{0} ∈ R_{1}^{3} if Σ(t) → χ_{0} as t → α + 0.

4.41. Lemma. Let M be a two-dimensional, doubly-connected, spacelike tube of zero mean curvature in R_{1}^{3} with a projection (0, β). Then ℳ can be defined by an immersion w = (x_{1}, x_{2}, t): K(1, R) → R_{1}^{3} of an annulus

K(1, R) = {ζ ∈ C : 1 < |ζ| < R}, ζ = ξ + iη,

such that

(4.42)  \begin{aligned}x_{1} & = \frac{μ}{4π} Re ∫_{ζ_{0}}^{ζ} \frac{1}{iz} (\frac{1}{g(z)} − g(z)) dz, \\ x_{2} & = \frac{μ}{4π} Re ∫_{ζ_{0}}^{ζ} \frac{1}{z}(\frac{1}{g(z)} + g(z)) dz, \\ t & = \frac{μ}{2π} log |ζ|, \end{aligned}

where g(z) is a holomorphic function on K(1, R) for which

(4.43)  Re ∮ \frac{i}{z} (\frac{1}{g} − g) dz = Re ∮ \frac{1}{z}(\frac{1}{g} + g) dz = 0.

Here R = e^{2πβ/μ} and

μ = ∫_{Σ(t)} |∇t|.

Proof. Because the function t(m) is harmonic with respect to the metric of ℳ by the Stokes formula, we conclude that the integral

∫_{Σ(t)} |∇t|

does not depend on t, that is, μ ≡ const, and the conjugate form ∗dt has a period

∫_{Σ(t)} ∗dt = μ.

There exists a multifunction h(m) such that dh = ∗dt, and the mapping

ζ(m) = exp \frac{2π}{μ} (t + i h)

establishes a one-to-one holomorphic correspondence between ℳ and K(1, R) (see [27]). By m = m(ζ) we denote the inverse mapping to ζ(m). Clearly,

t ∘ m = \frac{μ}{2π} log |ζ|.

Using the arguments of [37, Chapter 3, Section 3], we obtain a Weierstrass representation for spacelike surfaces of zero mean curvature in R_{1}^{3}

(4.44)  \begin{aligned}x_{1} & = \frac{1}{2} Re ∫_{ζ_{0}}^{ζ} f(z)(1 − g^{2}(z)) dz, \\ x_{2} & = \frac{1}{2} Re i ∫_{ζ_{0}}^{ζ} f(z)(1 + g^{2}(z)) dz, \\ t & = Re ∫_{ζ_{0}}^{ζ} if(z) g(z) dz, \end{aligned}

where f(z) and g(z) are holomorphic functions on K(1, R) satisfying the conditions

Re ∮ f(1 − g^{2}) dz = Re ∮ if(1 + g^{2}) dz = Re ∮ ifg dz = 0.

These conditions provide a tubular type of ℳ .

On the other hand,

Re ∫_{ζ_{0}}^{ζ} if(z)g(z) dz = \frac{μ}{2π} log |ζ|;

therefore, we can put

f(z) = \frac{μ}{2πizg(z)}.

Substituting the expression in (4.44), we obtain what is needed.

A similar result was obtained for minimal tubes in [35] and [36].

4.45. Lemma. Let ℳ ⊂ R_{1}^{3} be a spacelike tube of zero mean curvature having a singularity at the origin. Then it can be defined by an immersion (4.42) of the annulus K(1, e^{2πβ/μ}) with some holomorphic function g(z) such that

|g(z)| < 1, z ∈ K(1, e^{2πβ/μ}) and |g(e^{iϕ})| ≡ 1.

Proof. We note that ℳ is spacelike if and only if |g(z)| ≠ 1, z ∈ K(1, R). Below we shall suppose that |g(z)| < 1 because the substitution g(z) → 1/g(z) reflects the surface ℳ with respect to the plane x = 0. On the other hand, it is known [18] that ℳ is conelike in a neighborhood of the singularity, and also the totality of the tangent rays to ℳ at this point forms a light cone. This means that |g(z)| ≡ 1 on the interior circle of K(1, R).

Now it is sufficient to verify (4.43). In fact, if g(e^{iφ}) = e^{iθ(φ)} , then

Re \frac{1}{i} ∮ \frac{1}{z}(\frac{1}{g} − g) dz = Re \frac{1}{i} ∫_{0}^{2π} e^{−iφ}(e^{−iθ(φ)} − e^{iθ(φ)})ie^{iφ} dφ
= −2 Re i ∫_{0}^{2π} sin θ(φ) dφ = 0.

The second condition of (4.43) is verified similarly.

Let x_{1}(m) and x_{2}(m) be the coordinate functions of an immersion w(ζ): K(1, R) → R_{1}^{3} of a spacelike zero mean curvature tube with a projection (0, β). Both functions are harmonic with respect to the metric of the surface ℳ [21, Note 14]. Therefore, the Stokes formula implies that quantities

μ_{1} = ∫_{Σ(t)} ⟨∇x_{1}, ∇t⟩\frac{1}{|∇t|}, μ_{2} = ∫_{Σ(t)} ⟨∇x_{2}, ∇t⟩\frac{1}{|∇t|}

do not depend on t. According to this, we define the vector Q = μe_{0}+μ_{1}e_{1}+μ_{2}e_{2} , which will be called the flow vector of the tube ℳ [36].

We let K(ζ) denote the Gaussian curvature of ℳ = (K(1, R), w) at the point w(ζ).

4.46. Lemma. The Gaussian curvature K(s, t) of a conformal metric

dl^{2} = λ(s, t)(ds^{2} ± dt^{2})

is expressed by the following formula:

K(t, s) = −\frac{1}{2λ}[\frac{∂}{∂s}(\frac{λ_{s}}{λ} ) ± \frac{∂}{∂t}(\frac{λ_{t}}{λ} )].

Proof. We put E_{1} = ∂/∂s and E_{2} = ∂/∂t. Then

|E_{1}|^{2} = λ, |E_{2}|^{2} = ±λ and ⟨E_{1}, E_{2}⟩ = 0,

∇_{E_{1}}E_{2} − ∇_{E_{2}}E_{1} = \frac{∂^{2}}{∂s∂t} − \frac{∂^{2}}{∂t∂s} = 0.

It is known [6, Addition A] that

K(s, t) = ⟨R(e_{1}, e_{2})e_{2}, e_{1}⟩ = ±λ^{−2}⟨R(E_{1}, E_{2})E_{2}, E_{1}⟩,

where e_{i} = E_{i}/\sqrt{λ} and R( · , ·)· is the curvature tensor of the given metric. If the connection of the metric is denoted by ∇, then

R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z.

Using the well-known properties of connections, we obtain

\begin{aligned}⟨∇_{E_{2}}E_{2}, E_{2}⟩ & = \frac{1}{2}∇_{E_{2}}|E_{2}|^{2} = ±\frac{1}{2}λ_{t}, \\ ⟨∇_{E_{2}}E_{2}, E_{1}⟩ = −⟨E_{2}, ∇_{E_{2}}E_{1}⟩ & = ⟨E_{2}, ∇_{E_{1}}E_{2}⟩\frac{1}{2}∇_{E_{1}}|E_{2}|^{2} = ∓\frac{1}{2}λ_{s}. \end{aligned}

Therefore,

∇_{E_{2}}E_{2} = ∓\frac{λ_{s}}{2λ}E_{1} + \frac{λ_{t}}{2λ}E_{2}.

In the same way we find

∇_{E_{1}}E_{2} = \frac{λ_{t}}{2λ}E_{1} + \frac{λ_{s}}{2λ}E_{2}.

Now we get

⟨R(E_{1}, E_{2})E_{2}, E_{1}⟩ = ⟨∇_{E_{1}}∇_{E_{2}}E_{2}, E_{1}⟩ − ⟨∇_{E_{2}}∇_{E_{1}}E_{2}, E_{1}⟩

= ⟨∇_{E_{1}}(∓\frac{λ_{s}}{2λ}E_{1} + \frac{λ_{t}}{2λ}E_{2}), E_{1}⟩

− ⟨∇_{E_{2}}(\frac{λ_{t}}{2λ}E_{1} + \frac{λ_{s}}{2λ}E_{2}), E_{1}⟩

= ∓(\frac{λ_{s}}{2λ})′_{t}λ ∓ (\frac{λ_{s}}{2λ})⟨∇_{E_{1}}E_{1}, E_{1}⟩ + (\frac{λ_{t}}{2λ})

− (\frac{λ_{t}}{2λ})′_{t}λ − (\frac{λ_{t}}{2λ})⟨∇_{E_{2}}E_{1}, E_{1}⟩ − (\frac{λ_{s}}{2λ})⟨∇_{E_{2}}E_{2}, E_{1}⟩

= −\frac{−λ}{2}[±(\frac{λ_{s}}{2λ})′_{s} + (\frac{λ_{t}}{2λ})′_{t}],

because ⟨∇_{E_{1}}E_{1}, E_{1}⟩ = \frac{1}{2}λ_{s}. Therefore,

K(t, s) = −\frac{1}{2λ}[\frac{∂}{∂s}(\frac{λ_{s}}{λ} ) ± \frac{∂}{∂t}(\frac{λ_{t}}{λ} )],

and the lemma is proved.

Let ζ = ξ + iη. Then the Laplacian in coordinates (ξ, η) is denoted by the term 4∂^{2}/∂ζ∂ζ̄. From Lemma 4.46 it is not difficult to calculate

(4.47)  K(ζ) = 4\frac{∂^{2}}{∂ζ∂ζ̄}λ(ζ) = \frac{64π^{2}|g′(ζ)|^{2}|ζ|^{2}|g(ζ)|^{2}}{μ^{2}(1 − |g(ζ)|^{2})^{4}} ,

where from Lemma 4.45

λ(ζ) = \frac{μ^{2}}{16π^{2}|ζ|^{2}|g(ζ)|^{2}} (1 − |g(ζ)|^{2})^{2}.

We denote by sinh α(ζ) a sine of hyperbolic angle between the normal vector to the surface ℳ and the vector e_{0} . Using Lemma 4.45, we can write

sinh α(ζ) = \frac{2|g(ζ)|}{1 − |g(ζ)|^{2}} .

Below we show that it is convenient to describe a geometric structure of a tube in a neighborhood of an isolated singularity in terms of the quantity κ(ζ) = K(ζ)/ sinh^{4} α(ζ), called the specific curvature of the surface ℳ .

Using (4.47), we obtain

(4.48)  κ(ζ) = \frac{4π^{2}}{μ^{2}} |\frac{ζ}{g(ζ)}|^{2}|g′(ζ)|^{2}.

Our immediate aim is to show that the flow vector and the specific curvature of ℳ are characteristics of the first and second orders of a deviation of a tube from the light cone in a neighborhood of a singularity.

In order to accomplish this goal, we prove the following auxiliary statement about the asymptotic decomposition of the coordinate functions.

4.49. Lemma. Let ℳ be a spacelike zero mean curvature tube with a projection (0, β) defined by an immersion (4.44). We put

\begin{aligned}x_{1}(e^{iφ}) & = a_{0}(φ),  \frac{∂x_{1}}{∂r} (e^{iφ}) = a_{1}(φ), \\ x_{2}(e^{iφ}) & = b_{0}(φ),  \frac{∂x_{2}}{∂r} (e^{iφ}) = b_{1}(φ), \end{aligned}

and suppose that a_{0} , a_{1} , b_{0} , b_{1} are real analytic functions. Then

(4.50)

x_{1}(ζ) = ∑_{k=0}^{∞}(−1)^{k+1} \frac{1}{(2k)!}[a_{0}^{(2k)}(φ) − \frac{log r}{2k + 1}a_{1}^{(2k)}(φ)](log r)^{2k},

x_{2}(ζ) = ∑_{k=0}^{∞}(−1)^{k+1} \frac{1}{(2k)!}[b_{0}^{(2k)}(φ) − \frac{log r}{2k + 1}b_{1}^{(2k)}(φ)](log r)^{2k}.

258 V. A. Klyachin and V. M. Miklyukov If ℳ has a singularity, then

(4.51) a_{1}^{2}(φ) + b_{1}^{2}(φ) = \frac{μ^{2}}{4π^{2}} , a_{1}^{′2}(φ) + b_{1}^{′2}(φ) = \frac{μ^{2}}{4π^{2}} |g′(e^{iφ})|^{2},
(4.52)  ∫_{0}^{2π} a_{1}(φ) dφ = μ_{1},
(4.53)  ∫_{0}^{2π} b_{1}(φ) dφ = μ_{2}.

Proof. We put

x_{1}(ζ) = x_{1}(re^{iφ}) = ∑_{k=0}^{∞} a_{k}(φ)(log r)^{k}.

If ζ = ξ + i η and

D = (\frac{∂}{∂ξ} , \frac{∂}{∂η} ),

then

⟨Dφ, Dr⟩ = 0, Δ φ = 0, Δ log r = 0, |Dφ|^{2} = \frac{1}{r^{2}} .

Since the mapping w(ζ) is holomorphic, Δx(ζ) = 0. Therefore,

0 = Δx_{1}(ζ) = ∑_{k=0}^{∞}\{a′′_{k}(φ)\frac{1}{r^{2}} (log r)^{k} + a_{k}(φ)k(k − 1)\frac{1}{r^{2}} (log r)^{k−2}\}.

We get a system of differential equations

a′′_{k}(φ) + (k + 1)(k + 2)a_{k+2}(φ) = 0, k = 0, 1, 2, …,

which gives

a_{2k}(φ) = (−1)^{k+1}a_{0}^{(2k)}(φ)\frac{1}{(2k)!} and a_{2k+1}(φ) = (−1)^{k}a_{1}^{(2k)}(φ)\frac{1}{(2k + 1)!}.

We have obtained the necessary decomposition.

Further, we have

\begin{aligned}Dx_{1} & = \frac{μ}{4π} \overline{(\frac{1}{iζ} (\frac{1}{g(ζ)} − g(ζ)))} = \frac{μ}{4π} \frac{i}{ζ̄ḡ(ζ)} − \frac{μ}{4π} \frac{iḡ(ζ)}{ζ̄}, \\ Dx_{2} & = \frac{μ}{4π} \overline{(\frac{1}{ζ}(\frac{1}{g(ζ)} + g(ζ)))} = \frac{μ}{4π} \frac{1}{ζ̄ḡ(ζ)} + \frac{μ}{4π} \frac{ḡ(ζ)}{ζ̄}. \end{aligned}

Therefore,

−Dx_{1} + iDy_{1} = i\frac{μ}{2π} \frac{ḡ(ζ)}{ζ̄}.

On the other hand, as

Dx_{1}(e^{iφ}) = a_{1}(φ)e^{iφ}  and  Dx_{2}(e^{iφ}) = b_{1}(φ)e^{iφ},

we find

g(e^{iφ}) = −\frac{2πi}{μ}(a_{1}(φ) + ib_{1}(φ))  and  g′(e^{iφ}) = −\frac{2πi}{μ}(a′_{1}(φ) + ib′_{1}(φ)),

and we obtain (4.51).

In order to prove (4.52) and (4.53), we note the mapping w(ζ) is holomorphic and, consequently,

∫_{Σ(t)} ⟨∇x_{1}, ∇t⟩\frac{1}{|∇t|} = ∫_{u^{−1}∘Σ(t)} ⟨Dx_{1}, Dr⟩ = ∫_{0}^{2π} a_{1}(φ) dφ.

The equality (4.53) can be proved similarly.

We note that from (4.51), (4.52), (4.53) and Cauchy’s inequality we get

4.54. Corollary. We have

−μ^{2} + μ_{1}^{2} + μ_{2}^{2} ≤ 0.

This means that the flow vector Q = (μ, μ_{1}, μ_{2}) is not spacelike.

The following statement gives a geometric interpretation of coefficients at the decomposition of coordinate functions x_{1}(ζ) and x_{2}(ζ).

4.55. Lemma. Let ℳ ⊂ R_{1}^{3} be a spacelike tube of zero mean curvature having an isolated singularity at the origin. Then the function κ(ζ) has real analytic values κ(φ) on the unit circle ζ = e^{iφ}, 0 ≤ φ ≤ 2π, and also

(\frac{μ^{2}}{4π^{2}} )^{2}κ(φ) = −a_{1}(φ)a′′_{1}(φ) − b_{1}(φ)b′′_{1}(φ).

Proof. By (4.48), we have

κ(ζ) = \frac{4π^{2}}{μ^{2}} |\frac{ζ}{g(ζ)}|^{2}|g′(ζ)|^{2}.

As |g(e^{iφ})| ≡ 1, by the symmetry principle, g(ζ) can be holomorphically extended on the annulus

K(\frac{1}{R}, R) = \{ζ ∈ C : \frac{1}{R} < |ζ| < R\}, R = e^{2πβ/μ}.

Therefore, κ(φ) is a real analytic function, and from (4.51) it follows that

(4.56)  \begin{aligned}κ(φ) & = \frac{4π^{2}}{μ^{2}} |g′(e^{iφ})|^{2} = (a_{1}^{′2}(φ) + b_{1}^{′2}(φ))(\frac{4π^{2}}{μ^{2}} )^{2} \\ & = (\frac{4π^{2}}{μ^{2}} )^{2}(−a_{1}(φ)a′′_{1}(φ) − b_{1}(φ)b′′_{1}(φ)). \end{aligned}

Proof of Theorem 2.9. Let

h(m) = \frac{1}{2} log \frac{cosh α − 1}{cosh α + 1},

where cosh α is a cosine of the hyperbolic angle between the unit normal vector to ℳ and the time axis. Because the Gauss mapping of the surface ℳ is holomorphic, h(m) is harmonic with respect to the metric of ℳ , and also

lim_{f(m)→0} h(m) = 0.

Therefore, the sets

H_{τ} = {m : h(m) = τ}

are compact for small τ. The quantity

c = ∫_{H_{τ}} |∇h|

does not depend on τ, and

c = ∫_{Σ(t)} ⟨∇h, ∇t⟩\frac{1}{|∇t|}.

On the other hand, the last integral is a value of integral curvature of the curve Σ(t), which equals 2π. Therefore,

\begin{aligned}2π & = ∫_{Σ(τ)} ⟨∇h, ∇t⟩\frac{1}{|∇t|} ds \\ & = ∫_{\{(μ/2π) log |ζ|=τ\}} ⟨Dh, Dr⟩ = ∫_{\{(μ/2π) log |ζ|=0\}} |Dh|, \end{aligned}

that is,

(4.57)  ∫_{|ζ|=1} |g′(ζ)| = ∫_{0}^{2π} \sqrt{a′_{1}^{2}(φ) + b_{1}^{′2}(φ)} dφ = 2π.

As a_{1}^{2} + a_{2}^{2} ≡ μ^{2}/π^{2} , there exists a function θ(φ) such that

a_{1}(φ) = \frac{μ}{2π} cos θ(φ)  and  a_{2}(φ) = \frac{μ}{2π} sin θ(φ).

Then the equality (4.57) can be rewritten

(4.58)  ∫_{0}^{2π} |θ′(φ)| dφ = 2π.

We note that the representation (4.44) is invariant under rotations of the plane C. It means that the transformation of the variables ζ → e^{iα}ζ retains ℳ . By this fact and without losing generality, we will suppose that θ(0) = 0. By (4.58) the function θ is monotone on [0, 2π] and θ(2π) = 2π. Moreover, it is obvious that θ′(φ) > 0 by the maximum principle. Therefore,

\sqrt{κ(φ)} = θ′(φ)2π/μ.

Now let f(x, y) be a solution of (2.7) having an isolated singularity at (0, 0). We put x+iy = ϱe^{iψ}. As in Lemma 4.41 log r = 2πt/μ, the decomposition (2.10) follows directly from analyticity and monotonicity of θ(λ).

Next, we have

\frac{x(re^{iφ})^{2} + y(re^{iφ})^{2} − f^{2}(x(re^{iφ}), y(re^{iφ}))}{(x(re^{iφ})^{2} + y(re^{iφ})^{2})^{2}} = \frac{1}{3}κ((re^{iφ})) + o(log r).

We fix ψ, and suppose that φ satisfies ψ = θ(φ). By Lemma 4.49 we conclude that

ϱe^{iψ} = \frac{μ}{2π} log re^{iθ(φ)} + o(log r).

Therefore,

(4.59)  lim_{ϱ→0} κ^{∗}(ϱe^{iθ(φ)}) = κ(φ),

which implies real analyticity of the function κ^{∗}.

From the last equality and (4.56), we obtain

∫_{0}^{2π} \frac{dψ}{\sqrt{κ^{∗}(ψ)}} = ∫_{0}^{2π} \frac{θ′(φ) dφ}{\sqrt{κ^{∗}(θ(φ))}} = ∫_{0}^{2π} \frac{θ′(φ) dφ}{\sqrt{κ(φ)}} = μ.

By (4.56) it is not difficult to get the equality (2.13), from which it follows that

max_{[0,2π]} κ^{∗} ≥ \frac{4π^{2}}{μ^{2}}  and  min_{[0,2π]} κ^{∗} ≤ \frac{4π^{2}}{μ^{2}} .

The theorem is proved.

The characteristic κ^{∗} introduced for the behavior of a solution of the spacelike zero mean curvature surfaces equation in the neighborhood of the singularity is complete.

4.60. Theorem. Let f_{1}(x, y) and f_{2}(x, y) be two solutions of (2.7) defined in a neighborhood of its common isolated singular point (0, 0). We suppose that the limit values

κ_{i}^{∗}(ψ) = 6 lim_{ϱ→0} \frac{ϱ − f_{i}(ϱe^{iψ})}{ϱ^{3}}, i = 1, 2,

of these solutions are equal. Then f_{1}(x, y) ≡ f_{2}(x, y).

Proof. By (4.59) and the equivalence κ_{1}^{∗} ≡ κ_{2}^{∗}, we can conclude that θ′_{1}(φ) ≡ θ′_{2}(φ). Hence, θ_{1}(φ) ≡ θ_{2}(φ). Therefore, g_{1}(e^{iφ}) ≡ g_{2}(e^{iφ}). Using the uniqueness theorem for holomorphic functions, we obtain g_{1}(ζ) ≡ g_{2}(ζ). The representation (4.44) leads to the equality f_{1} ≡ f_{2} .

5. Timelike surfaces

In this section we investigate timelike tubular surfaces of zero mean curvature in a neighborhood of a singular point. Below we will suppose that the Cartesian coordinates (x_{1}, …, x_{n}, t) ∈ R_{1}^{n+1} are determined so that the scalar square of a vector χ = (x_{1}, …, x_{n}, t) is expressed by

|χ|^{2} = −t^{2} + ∑_{i=1}^{n} x_{i}^{2}.

Let M be a two-dimensional connected orientable C^{4} -manifold without a boundary. We consider a surface ℳ = (M, u) defined by a C^{3} -immersion u: M → R_{1}^{n+1} . A surface ℳ is called timelike if each of its tangent planes contains timelike vectors. Because the metric of timelike surfaces is indefinite, we introduce the norm ∥ · ∥ by

∥X∥ = \sqrt{|⟨X, X⟩|} .

5.61. Lemma. Let ℳ be a two-dimensional timelike tube of zero mean curvature in R_{1}^{n+1} . Then,

(a) with respect to the metric of the surface ℳ , Δx_{i} = 0, i = 0, 1, …, n;

(b) the quantity μ = ∫_{Σ(t)} ∥∇t∥ does not depend on t;

(c) if (t, s) are local isothermal coordinates on ℳ , then

\frac{∂^{2}x_{i}}{∂t^{2}} = \frac{∂^{2}x_{i}}{∂s^{2}} for any i = 0, 1, …, n.

Proof. The harmonicity of the coordinate functions of ℳ is known as well as the harmonicity of the coordinate functions of minimal surfaces in the Euclidean space [21, Note 14].

Using the Stokes formula with Δt = 0 as well as the above one, we conclude that μ is independent of t.

The third statement follows from (a) and from the special expression of the Laplacian in local isothermal coordinates.

In fact, if the metric is dl^{2} = λ^{2}(ds^{2} − dt^{2}) and f(s, t) is a C^{2} -function, then

∇f = (\frac{1}{λ^{2}} f_{s}, −\frac{1}{λ^{2}} f_{t})

and

Δf = div (∇f) = ⟨∇_{E_{1}}∇f, E_{1}⟩ − ⟨∇_{E_{2}}∇f, E_{2}⟩ = \frac{1}{λ^{2}} (f_{ss} − f_{tt}),

where E_{1} = λ^{−1}∂/∂s and E_{2} = λ^{−1}∂/∂t.

We will prove the following auxiliary statement.

5.62. Lemma. Let ℳ ⊂ R_{1}^{n+1} be a two-dimensional doubly-connected timelike C^{2} -tube of zero mean curvature with a projection (α, β) and a flow μ. Then ℳ can be represented by a C^{2} -immersion

u(t, s) = \frac{1}{2}(r(s + t) + r(s − t)) + \frac{1}{2} ∫_{s−t}^{s+t} h(λ) dλ + e_{0}t,

of the string (α, β) × (−∞, +∞).

Here r, h: R → R^{n} are vector functions such that for any s ∈ R

(5.63)  |r′(s)|^{2} + |h(s)|^{2} = 1,  ⟨r′(s), h(s)⟩ = 0,

and

(5.64)  r′(s + μ) + h(s + μ) = r′(s) + h(s).

Proof. With respect to the metric of ℳ , the function t(m) satisfies the following differential equation:

Δt = div ∇t = 0.

Therefore, the differential form ∗dt is closed. Hence, there exists a multifunction δ(m) such that dδ = ∗dt and

∫_{Σ(t)} dδ = μ = ∫_{Σ(t)} ∥∇t∥.

We consider the multivalued mapping

F: M → (α, β) × (−∞, +∞), where m → (t(m), δ(m)).

Note that u(t, s) is μ-periodic by the variable s.

We denote the inverse mapping to F by u(t, s). We assume that

\overline{R}(t, s) = u(t, s) − e_{0}t.

We note that the coordinates (t, s) are isothermal on ℳ . Because the mean curvature is equal to zero by Lemma 5.61, we conclude

\frac{∂^{2}\overline{R}}{∂t^{2}} = \frac{∂^{2}\overline{R}}{∂s^{2}}.

Using d’Alambert’s formula [38, Section 13], we get

(5.65)  u(t, s) = \frac{1}{2}(r(s + t) + r(s − t)) + \frac{1}{2} ∫_{s−t}^{s+t} h(λ) dλ + e_{0}t,

where r, h: R → R^{n} are some vector functions. Since the coordinates (t, s) are isothermal,

|u_{t}|^{2} = −|u_{s}|^{2}, ⟨u_{t}, u_{s}⟩ = 0.

For vector functions r(s) and h(s), these equalities can be rewritten in the form

\begin{aligned}|r′(s + t) + h(s + t)|^{2} + |r′(s − t) − h(s − t)|^{2} & = 2, \\ |r′(s + t) + h(s + t)|^{2} − |r′(s − t) − h(s − t)|^{2} & = 0, \end{aligned}

which is the same as (5.63) and (5.64).

From (5.65) we conclude that if and only if the function r′(s) + h(s) is μperiodic, ℳ is a tube. The lemma is proved.

Below we give a representation of a timelike surface of zero mean curvature having a singularity in R_{1}^{3} .

5.66. Theorem. Let ℳ ⊂ R_{1}^{3} be a two-dimensional doubly-connected timelike C^{2} -tube of zero mean curvature with projection (0, β). Then there exists a C^{1} -function θ: (−∞, +∞) → R such that for some integer k, it is true that θ(s + μ) = θ(s) + 2πk, and ℳ can be represented in the form

(5.67)  \begin{aligned}x(τ, s) & = \frac{1}{2} ∫_{s−τ}^{s+τ} cos θ(λ) dλ; \\ y(τ, s) & = \frac{1}{2} ∫_{s−τ}^{s+τ} sin θ(λ) dλ; \\ t & = τ. \end{aligned}

Geometric structure of tubes and bands of zero mean curvature 265 Proof. Because ℳ has a singularity at the origin, we have r(s) ≡ 0. From

Lemma 5.62, we obtain |h(s)| ≡ 1 and h(s + μ) = h(s).

Therefore, there exists a function θ(λ) such that

h(λ) = (cos θ(λ), sin θ(λ)), θ(λ + μ) = θ(λ) + 2πk, k = 0, 1, 2, …. We denote the Gaussian curvature by K(s, t) and by cosh α a hyperbolic

cosine of an angle between a unit normal to ℳ and the time axis. The quantity

κ = \frac{K}{cosh^{4} α}

is said to be a specific curvature of ℳ .

5.68. Lemma. The following formulas are true:

\begin{aligned}cosh^{2} α & = (sin^{2} \frac{θ(s + t) − θ(s − t)}{2})^{−1}, \\ κ(t, s) & = θ′(s + t)θ′(s − t). \end{aligned}

Proof. By Theorem 5.66, the metric of ℳ has the form

ds_{ℳ}^{2} = sin^{2} ω(ds^{2} − dt^{2}), where ω = \frac{1}{2}(θ(s + t) − θ(s − t)). Therefore,

cosh^{2} α = ∥e_{0}^{T} ∥^{2} = ∥∇t∥^{2} = \frac{1}{sin^{2} ω} .

From Lemma 4.46, we have

K(s, t) = −\frac{1}{sin^{2} ω} Δ′ log sin ω = −\frac{1}{sin^{2} ω} (\frac{Δ′ sin ω}{sin ω} − \frac{|D sin ω|^{2}}{sin^{2} ω} ), where

Δ′ = \frac{∂^{2}}{∂s^{2}} − \frac{∂^{2}}{∂t^{2}} , D = (\frac{∂}{∂s}, −\frac{∂}{∂t}).

Since

|Dω|^{2} = (\frac{∂ω}{∂s} )^{2} − (\frac{∂ω}{∂t} )^{2} and Δ′ω = 0,

the Gaussian curvature can be rewritten as

K(s, t) = \frac{|Dω|^{2}}{sin^{4} ω} .

Hence, the specific curvature is expressed by the equality

κ(s, t) = |Dω|^{2} = θ′(s + t)θ′(s − t).

Proof of Theorem 2.15. Let Σ(τ) be a section of ℳ by the plane t = τ. We note that the curve Σ(τ) is closed and can be defined by

x(τ, s) = \frac{1}{2} ∫_{s−τ}^{s+τ} cos θ(λ) dλ  and  y(τ, s) = \frac{1}{2} ∫_{s−τ}^{s+τ} sin θ(λ) dλ.

We denote the curvature of this curve by k(s). Using the formula

k(s) = \frac{x′y′′ − y′x′′}{(x^{′2} + y^{′2})^{3/2}} ,

we get

k(s) = \frac{θ′(s + t) + θ′(s − t)}{2|sin ω|}.

Because the length element of Σ(t) is dl = |sin ω| ds, we can write

∫_{Σ(τ)} k(s) dl = \frac{1}{2} ∫_{0}^{μ} θ′(s + τ) ds + \frac{1}{2} ∫_{0}^{μ} θ′(s − τ) ds.

On the other hand, as Σ(τ) is closed,

∫_{Σ(τ)} k(s) dl = 2π,

and, therefore,

2π = ∫_{0}^{μ} θ′(s) ds = θ(μ) − θ(0).

Hence, we conclude that the range of θ(s) is [0, 2π].

We put

ξ_{τ} = (\frac{1}{2}(cos θ(s + τ) + cos θ(s − τ)), \frac{1}{2}(sin θ(s + τ) + sin θ(s − τ)), 1).

Clearly, the vectors ξ_{τ} are orthogonal to Σ(τ) and tangent to ℳ . Let τ → 0. We find that the totality of the vectors

ξ_{0} = (cos θ(s), sin θ(s), 1)

forms a light cone. This follows from the above properties of θ(s).

We put ψ = θ(s).

5.69. Theorem. Let f(x, y) be a solution of (2.14) having an isolated singularity at (0, 0) with f(0, 0) = 0. Then

lim_{ϱ→0} \frac{f(ϱe^{iψ}) − ϱ}{ϱ^{3}} = \frac{1}{6}κ(s)  and  ϱe^{iψ} = x + iy.

Proof. We have

x′_{t}(s, t) = \frac{1}{2}(cos θ(s + t) + cos θ(s − t)),
x′′_{tt}(s, t) = \frac{1}{2}(−θ′(s + t) sin θ(s + t) + θ′(s − t) sin θ(s − t)),
y′_{t}(s, t) = \frac{1}{2}(sin θ(s + t) + sin θ(s − t)),
y′′_{tt}(s, t) = \frac{1}{2}(θ′(s + t) cos θ(s + t) − θ′(s − t) cos θ(s − t)).

Computing directly, we get

lim_{t→0} \frac{x_{t}^{′2} + y_{t}^{′2} − 1}{t^{2}} = −θ^{′2}(s)  and  lim_{t→0} \frac{x x′′ + y y′′}{t} = −θ^{′2}(s).

Using Lemma 5.68, we find that

lim_{t→0} \frac{t − \sqrt{x^{2} + y^{2}}}{(\sqrt{x^{2} + y^{2}} )^{3}} = \frac{κ(s)}{6}.

6. Smooth pasting

In this section, we investigate the possibility of a smooth pasting of spacelike and timelike tubes in a neighborhood of their common singular point. To begin, we prove the following auxiliary statement.

6.70. Lemma. Let f(x, y) be a solution of (2.14) with an isolated singularity (0, 0). We assume that the function θ(λ) in (5.67) is real analytic and satisfies θ′(λ) > 0. Then there exist real analytic functions h_{k}(ψ), k = 1, 2, …, defined on [0, 2π] so that

(6.71)  f(ϱe^{iψ}) = ϱ + ∑_{k=1}^{∞} h_{k}(ψ)ϱ^{2k+1}.

Proof. We consider the mapping (ϱ, ψ) of the rectangle

Q = (0, μ) × (0, β)

in R^{2} , defined by

(6.72)  ϱ = \frac{1}{2}((∫_{s−t}^{s+t} cos θ(λ) dλ)^{2} + (∫_{s−t}^{s+t} sin θ(λ) dλ)^{2})^{1/2}

and

(6.73)  ψ = arctg (∫_{s−t}^{s+t} cos θ(λ) dλ)(∫_{s−t}^{s+t} sin θ(λ) dλ)^{−1}.

For the Jacobian J(s, t) of this mapping, we have

J(s, 0) = θ′(s) > 0.

We denote by G(ϱ, ψ) the local inverse mapping to (6.72) and (6.73). As θ(s) is real analytic, the mapping G is also real analytic. Differentiating (6.72) directly, we see that

\frac{∂^{2k}t}{∂ϱ^{2k}} |_{ϱ=0}= 0.

Therefore, there exist real analytic functions h_{k}(ψ), k = 1, 2, …, such that

t = ϱ + ∑_{k=1}^{∞} h_{k}(ψ)ϱ^{2k+1},

that is, the decomposition (6.71) holds.

Proof of Theorem 2.16. Let f_{1}(x, y) be a solution of (2.7) with singularity at (0, 0) and a flow μ. Using (2.10), we obtain

f_{1}(ϱe^{iψ}) = ϱ − \frac{1}{6}κ^{∗}(ψ)ϱ^{3} + ∑_{k=2}^{∞} c_{k}(ψ)ϱ^{2k+1}.

Consider a solution of (2.14) constructed by (5.67) with

θ(s) = ∫_{0}^{2πs/μ} \sqrt{κ(λ)} dλ.

By (6.71) we find

f_{2}(ϱe^{iψ}) = −ϱ − \frac{1}{6}κ^{∗}(ψ)ϱ^{3} + ∑_{k=2}^{∞} h_{k}(ψ)ϱ^{2k+1}.

Using the analyticity of the functions c_{k}(ψ) and h_{k}(ψ), we conclude that the function δ(ϱe^{iψ}) ∈ C^{2} in some neighborhood of (0, 0). The theorem is proved.

Geometric structure of tubes and bands of zero mean curvature 269

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Received 10 October 2001