Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 227–236

LENGTHS OF GEODESICS ON RIEMANN SURFACES WITH BOUNDARY

Hugo Parlier

Universidad Nacional de Educación a Distancia, Facultad de Ciencias Departamento de Matemáticas Fundamentales, ES-28040 Madrid, Spain hugo.parlier@epfl.ch

Abstract. This article concerns lengths of simple closed geodesics on hyperbolic Riemann surfaces. In particular, for a surface with boundary, it is shown that boundary length can be increased such that all simple closed geodesics are lengthened.

1. Introduction

Consider a Riemann surface S endowed with a hyperbolic metric. The surface’s simple length spectrum Δ_{0}(S) is the ordered set of lengths {l_{1} ≤ l_{2} ≤ ⋯} of its (interior) simple closed geodesics. The length l_{1} is its systole length (the shortest non-trivial closed curve) and l_{1} is known to be bounded for any given genus if S is a closed surface (without boundary). Surfaces that satisfy this bound have been a subject of active research (i.e. [6], [9]). This is perhaps the best known example of the larger study of extremal surfaces for the length of a certain simple closed geodesic. The idea of this article is to prove the following theorem, which is a natural tool for describing extremal surfaces.

Theorem 1.1. Let S be a compact hyperbolic Riemann surface with nonempty boundary and let ε > 0. There exists a surface S̃ of same signature with total boundary length ε greater than that of S with Δ_{0}(S) < Δ_{0}(S̃).

In other words, boundary length can be increased so that a simple closed geodesic on S is strictly shorter that its corresponding geodesic on S̃. The surfaces S and S̃ are of same topological type endowed with two different hyperbolic metrics. The reciprocal to the theorem, namely that boundary length can be decreased in order to decrease the length of all simple closed geodesics, is also true. It must noted that a similar proposition limited to the case where S is of signature (g, 1), was proved by Schmutz–Schaller in [9] and was used to find properties for surfaces with maximum size systoles.

The theorem, along with the convexity of geodesic length functions along earthquake paths, is used to prove the following corollaries.

2000 Mathematics Subject Classification: Primary 30F10; Secondary 30F45.

The author was supported by the Swiss National Science Foundation grants 21-57251.99 and 20-68181.02.

Corollary 1.2. Let S be a maximal surface of genus g for Bers’ constant. Then all simple closed geodesics intersect at least two distinct simple closed geodesics of length L_{g} that are the longest geodesics of distinct partitions.

Corollary 1.3. Let S be a surface of genus g with largest possible systole among all surfaces of same genus. Let γ be a simple closed geodesic. Then γ intersects two distinct systoles (and distinct from γ).

Acknowledgements. I would like to thank the referee for helpful comments, and Peter Buser for his advice, suggestions and encouragement that have made this work possible. I would also like to thank Aline Aigon-Dupuy for her help with the figures.

2. Preliminaries

Here a surface will always be a compact Riemann surface equipped with a metric of constant curvature −1. Such a surface is always locally isometric to the hyperbolic plane H. A surface will generally be represented by S and distance on S (between points, curves or other subsets) by d_{S}( · , ·). The signature of such a surface will be denoted (g, n) (genus g with n boundary curves). All boundary curves must be smooth closed geodesics. A surface of signature (0, 3) is called a Y -piece or a pair of pants and will generally be represented by 𝒴 or 𝒴_{i}.

A curve, unless mentioned, will always be non-oriented and primitive. The set of all free homotopy classes of closed curves of a surface S is denoted by π(S). A non-trivial curve on S is a curve which is not freely homotopic to a point. A closed curve on S is called simple if it has no self-intersections. Closed curves (geodesic or not) will generally be represented by greek letters (α, β, γ and γ_{i} etc.) whereas paths (geodesic or not) will generally be represented by lower case letters (a, b etc). The intersection number between two distinct curves α and β will be denoted int(α, β). Unless otherwise specified, a geodesic is a simple closed geodesic curve. A non-separating closed curve is a closed curve γ such that the set S \ γ is connected. Otherwise, a closed curve is called separating. The function that associates to a finite path or curve its length will be represented by l( ·), although generally a path or a curve’s name and its length will not be distinguished.

The simple length spectrum of a surface S is the ordered set of lengths of simple closed geodesics and will be denoted

Δ_{0}(S) = {l_{1} ≤ l_{2} ≤ ⋯}.

By convention, for S with boundary only interior the lengths of interior simple closed geodesics will be allowed to appear in Δ_{0}(S). Consider two surfaces S and S̃ with simple length spectrums Δ_{0}(S) = {l_{1} ≤ l_{2} ≤ ⋯} and Δ_{0}(S̃) = {l̃_{1} ≤ l̃_{2} ≤ ⋯}. The notation Δ_{0}(S) < Δ_{0}(S̃) is an abbreviation for l_{i} < l̃_{i} for all i ∈ N^{∗}.

A geodesic length function is an application that associates length to geodesics according to homotopy class under the action of a continuous transformation of a surface. In [8], S. Kerckhoff proves the following theorem.

Theorem 2.1. The geodesic length function of a simple closed curve δ is convex along earthquake paths. It is strictly convex if δ intersects γ, the simple closed geodesic along which the earthquake was performed.

A surface of genus g can always be cut along 3g − 3 disjoint simple closed geodesics {γ_{1}, …, γ_{3g−3}} which separate S into 2g − 2 Y -pieces. The reunion of these geodesics 𝒫 = {γ_{1}, …, γ_{3g−3}} is called a partition. (If S is of signature (g, n) then a partition contains 3g −3+n simple closed geodesics.) If 𝒫 is chosen such that max_{i=1}^{3g−3} l(γ_{i}) = l(𝒫) is minimal among all possible partitions of S, then l(𝒫) ≤ L_{g}, where L_{g} is constant depending uniquely on the genus ([1], [2]). The optimal constant L_{g} is often called the Bers’ partition constant and the best known bound is L_{g} ≤ 21(g − 1) ([3, Remark 5.2.5, p. 129]). A surface of genus g will be called maximal for Bers’ partition constant if it is a surface on which a minimal partition 𝒫 satisfies l(𝒫) = L_{g}.

A Y -piece can be separated into two symmetric isometric right angled hyperbolic hexagons. To do this, one can cut along the three disjoint simple geodesic paths that go between the boundary geodesics. The following list of well-known propositions concerning hyperbolic polygons (polygons in the hyperbolic plane H) are of particular use in the proofs, and are included for sake of completeness. They can be found in either [3] or [5]. All polygons, unless specially mentioned, are considered right-angled.

Proposition 2.2 ([3, Theorem 2.4.1, p. 40]). Let H be a hexagon with a, b and c non-adjacent sides. Let α be the remaining edge adjacent to b and c, β the remaining edge adjacent to a and c, and γ the remaining edge adjacent to a and b. Then

cosh c = sinh a sinh b cosh γ − cosh a cosh b. Proposition 2.3 ([3, Theorem 2.4.4, p. 42]). Let H be a non-convex hexagon

with a, b and c non-adjacent sides. Let α be the remaining edge adjacent to b and c, β the remaining edge adjacent to a and c, and γ the remaining edge adjacent to a and b. Let H be such that γ and c intersect. Then

cosh c = sinh a sinh b cosh γ + cosh a cosh b.

A trirectangle is a quadrilateral with three right angles.

Proposition 2.4 ([3, Theorem 2.3.1, p. 37]). Let R be a trirectangle with interior angle φ being the only non-right angle situated between sides α and β. Let a and b be the remaining sides with a adjacent to β and b adjacent to α. Then the following formulas hold:

cos φ = sinh a sinh b and sinh α = sinh a cosh β.

The following proposition deals with quadrilaterals with only two right angles.

Proposition 2.5 ([3, Theorem 2.3.1, p. 38]). Let R be a convex quadrilateral with two right interior angles. Let γ be the side of R between the two right angles. Let c be the side opposite γ and a and b the remaining sides. Then

cosh c = cosh a cosh b cosh γ − sinh a sinh b.

And in the non-convex case the following proposition holds.

Proposition 2.6 ([3, Theorem 2.3.1, p. 38]). Let R be a non-convex quadrilateral with two right interior angles. Let γ be the side of R between the two right angles and c be the side that intersects γ. Let a and b be the remaining sides. Then

cosh c = cosh a cosh b cosh γ + sinh a sinh b.

Finally:

Proposition 2.7 ( [3, Example 2.2.7, p. 36]). Let P be a pentagon with four right angles. Let φ be the (only non-right) interior angle between two sides, a and b. Let α be the other side adjacent to b and β the other side adjacent to a. Let c be the remaining edge. Then the following formulas hold:

cosh c = − cosh a cosh b cos φ + sinh a sinh b and \frac{cosh a}{cosh α} = \frac{cosh b}{cosh β} = \frac{cosh c}{cosh φ}.

3. Main theorem and corollaries

The proof of the main theorem is primarily based on the following technical lemma. The lemma essentially states that by decreasing a boundary length on a Y -piece, one decreases the length of all simple geodesic paths between the other boundary geodesics.

Lemma 3.1. Let 𝒴 = (α, β, γ) be a Y -piece. Let d_{αβ}, d_{αγ} and d_{βγ} be 𝒴 ’s three perpendiculars. Let c be a border to border simple geodesic path from α to β (from α to α, respectively). Let p and q be the initial and end points of c. For a positive ε < l(γ) let 𝒴 ′ = (α, β, γ′) with l(γ′) = l(γ) − ε. Let p′ be the point on α ⊂ 𝒴 ′ corresponding to p (such that d_{𝒴 ′} (p′, d_{αγ′}) = d_{𝒴} (p, d_{αγ}) and d_{𝒴 ′} (p′, d_{αβ}) = d_{𝒴} (p, d_{αβ})) and q′ the point on β ⊂ 𝒴 ′ corresponding to q. Let c′ be the simple geodesic path on 𝒴 ′ from p′ to q′. Then c′ < c.

Proof. The three arcs d_{αβ}, d_{αγ} and d_{βγ} separate 𝒴 into two right-angled hexagons. The geodesic path c can either cross one or both hexagons. This explains the different configurations of c in what follows. For what follows also notice that if the length of γ decreases, then so does d_{αβ} (Proposition 2.2).

Let c be a simple arc on 𝒴 from α to β. Then one the two following figures hold.

Figure 1. A Y -piece with a geodesic arc from α to β.

Figure 2. Possibility 1 for the path c.

Consider the quadrilaterals containing c and d_{αβ} (convex in Figure 2 and non-convex in Figure 3). The paths (and their lengths) d_{p} and d_{q} are defined as the two remaining sides of these quadrilaterals, as in the figures. (The path d_{p} is between p and d_{αβ}, and the path d_{q} is between q and d_{αβ}.) Notice that according to the hypotheses these lengths do not change under the influence of the modification to 𝒴 . Using the formula for quadrilaterals (Proposition 2.5), in Figure 2 we have the following expression for the length of c:

cosh c = cosh d_{p} cosh d_{q} cosh d_{αβ} − sinh d_{p} sinh d_{q}.

In Figure 3, the quadrilateral with sides d_{p}, d_{q}, c and d_{αβ} is the situation described in Proposition 2.6. Thus:

cosh c = cosh d_{p} cosh d_{q} cosh d_{αβ} + sinh d_{p} sinh d_{q}.

Figure 3. Possibility 2 for the path c.

Figure 4. A Y -piece with a geodesic arc from α to α.

In both cases it is easy to see that the length of c is strictly reduced by a reduction of length γ.

Let c be a simple arc on 𝒴 from α to α. The paths d_{p} and d_{q} are defined as previously and verify the configuration of either Figures 5 or 6. In both cases we can extract the hexagon depicted in Figure 7.

Notice that h, the common perpendicular between c and β is contained in the hexagon as in Figure 7. β_{1} and β_{2} are defined as the two parts of β separated by h. Notice that the pentagons (Q_{1} ∪Q_{2}) and (Q_{1} ∪Q_{2}) are right-angled, except in q and p, respectively. Thus, using Proposition 2.7 the following formulas hold:

\frac{cosh c_{1}}{sinh d_{αβ}} = \frac{cosh d_{q}}{sinh h} and \frac{cosh c_{2}}{sinh d_{αβ}} = \frac{cosh d_{p}}{sinh h} .

Figure 5. Possibility 1 for the path c.

Figure 6. Possibility 2 for the path c.

Figure 7. Hexagon used to calculate the variation of c.

Thus

(1)  cosh c_{1} = \frac{cosh d_{q}}{cosh d_{q}} cosh c_{2}.

From this formula, and because c = c_{1} + c_{2} , if either c_{1} or c_{2} decreases then so does c.

When there is a variation of γ then, because β is constant, either β_{1} or β_{2} is reduced (or stays constant). The situation being symmetric, suppose that β_{1} is reduced or left constant.

We can now use the trirectangle Q_{1} and Proposition 2.4. In Figure 7 the following geodesic paths need to be defined: h_{q} is the minimal path from q to β, h_{p} is the minimal path from p to β, and β_{11} and β_{12} are the two parts of β_{1} separated by h_{q}. The following formula holds:

sinh h_{q} = sinh d_{αβ} cosh d_{q}.

Thus h_{q} is reduced by a reduction of γ. In the same trirectangle we can see that

sinh β_{11} = \frac{sinh d_{q}}{cosh h_{q}} .

This proves that β_{11} increases, and this implies that β_{12} = β_{1}−β_{11} decreases. Now in the trirectangle Q_{2} the following formula holds:

sinh c_{1} = sinh β_{12} cosh h_{q}.

Both β_{11} and h_{q} decrease and this implies that c_{1} decreases as well. We can now conclude that c decreases as well.

Theorem 3.2. Let S be a surface of signature (g, n) with n > 0. Let γ_{1}, …, γ_{n} be the boundary geodesics of S. For (ε_{1}, …, ε_{n}) ∈ (R^{+})^{n} with at least one ε_{i} ≠ 0, there exists a surface S̃ with boundary geodesics of length γ_{1} + ε_{1}, …, γ_{n} + ε_{n} such that all corresponding simple closed geodesics in S̃ are of length strictly greater than those of S (Δ_{0}(S) < Δ_{0}(S̃)).

Proof. For a given partition of S, a simple closed geodesic is either an element of the partition, or transversally intersects the partition. In the latter case, it can be seen as the union of simple geodesic arcs, each arc joining geodesics in the partition. The previous lemma will thus be our main tool for comparing lengths of simple geodesics between surfaces.

Let 𝒫 be a partition of S. We shall replace S by the following surface S̃. Let γ_{i} be a boundary geodesic and 𝒴_{γ_{i}}(α, β, γ_{i}) be the element of 𝒫 with γ_{i} a boundary geodesic. For ε > 0, replace 𝒴_{γ_{i}} with 𝒴 = (α, β, γ_{i} + ε) without modifying the twist parameters.

Let us denote 𝒫̃ the partition on S̃ corresponding to 𝒫. Now let δ̃ be a simple closed geodesic on S̃. We shall now find a corresponding geodesic δ on S. To find δ, proceed as follows. The geodesic δ̃ is either an element of 𝒫̃, or intersects the partition. If we are in the latter case, then δ̃ = ⋃_{i=1}^{m} c̃_{i} where each c̃_{i} is a simple geodesic path from an element of 𝒫̃ to another. To each c̃_{i} one can find a corresponding c_{i} on S as in the previous lemma. Now let δ = 𝒢 (⋃_{i=1}^{m} c_{i}). Clearly, by the previous lemma, if δ intersects 𝒴_{γ_{i}}, then l(δ) < l(δ̃).

The same process can be performed by increasing the length of multiple boundary geodesics simultaneously. Notice that all simple closed geodesics on S are in one-to-one correspondence with those of S̃. For a given partition 𝒫 on S, all simple closed geodesics that have a transversal intersection with a Y -piece whose boundary length has been increased correspond to a simple closed geodesic on S̃ whose length is strictly greater. Conversely, all other simple closed geodesics on S correspond to simple closed geodesics on S̃ with identical length.

To construct S̃, proceed as follows. Let (ε̃_{1}, …, ε̃_{1})) = (ε_{1}/k, …, ε_{n}/k) where k is a positive integer to be determined. Take a partition 𝒫 on S, and construct a surface S_{1} as described above with boundary geodesics of lengths γ_{1} +ε̃_{1}, …, γ_{n} + ε̃_{n}. On S_{1} , choose a different partition 𝒫_{1} and repeat the process to obtain a surface S_{2} etc. For each step, as the correspondence between geodesics is one-to-one, there continues to be a one-to-one correspondence between geodesics on S and geodesics on, say, S_{m}. Clearly, by choosing the partitions 𝒫_{m} correctly, it takes a finite number of steps N so that the length of all simple closed geodesics on S are strictly inferior to their corresponding geodesics on S_{N}. The number N depends on the genus and the number of boundary geodesics γ_{i} whose length is increased in the process. Now replace k by N and the corresponding construction proves the theorem.

Notice that it suffices to have one boundary geodesic whose length increases.

The idea of studying maximal surfaces has induced us to focus on how to increase the length of simple closed geodesics. However, the above proof also applies to the reduction of lengths, and the following theorem holds as well.

Theorem 3.3. Let S be a surface of signature (g, n) with n > 0. Let γ_{1}, …, γ_{n} be the boundary geodesics of S. For (ε_{1}, …, ε_{n}) ∈ (R^{−})^{n} with at least one ε_{i} ≠ 0, there exists a surface S̃ with boundary geodesics of length γ_{1} + ε_{1}, …, γ_{n} + ε_{n} such that all corresponding simple closed geodesics in S̃ are of length strictly less than those of S (Δ_{0}(S) > Δ_{0}(S̃)).

As a first corollary to Theorem 3.2, let us first look at properties of maximal surfaces for Bers’ partition constant.

Corollary 3.4. Let S be a maximal surface of genus g for Bers’ constant. Then all simple closed geodesics intersect at least two distinct simple closed geodesics of length L_{g} that are the longest geodesics of distinct partitions.

Proof. Let γ be a non-separating simple closed geodesic on S. Open S along γ to obtain a surface S̃ of signature (g − 1, 2). By the previous proposition, S̃ can be replaced by a new surface S′ with all lengths of simple closed geodesics longer, including γ. If S was maximal for genus g, then the surface obtained by gluing S′ back together again along the image of γ must have a shorter partition. The only geodesics that could have been shortened by this operation were those that intersected γ, thus γ must have intersected a geodesic δ that was of length L_{g} and that completed a partition as the longest geodesic. Now twist γ to obtain a new surface S′′ such that the image of δ is longer. Re-perform the operation described above and S can only have been maximal if there was a second distinct geodesic like δ, which proves the result.

236 Hugo Parlier

Remark 3.5. An upper bound on L_{g} is known ([4], [3]). On a maximal surface the shortest simple closed geodesic σ intersects a geodesic of length L_{g}. With this in mind, using the collar theorem [7], we can give a lower bound on σ (which is not sharp) for a maximal surface in the following manner. The collar theorem states that there is a lower bound 2 arcsinh(2/σ) for the length of any closed geodesic that intersects σ. Thus for σ too small, we have

2 arcsinh(2/σ) > L_{g},

and thus the surface in question cannot be maximal. Applied differently, the same argument shows that surfaces with small systoles admit a partition containing the systoles and with all geodesics of length inferior to L_{g}.

A second corollary, similar to the first one, concerns surfaces with maximum size systoles. The problem of maximum size systoles is exposed in [9], and a sharp bound is known for surfaces of genus 2 ([6]). The proof of this corollary is almost identical to the previous corollary and is left to the reader.

Corollary 3.6. Let S be a surface of genus g with largest possible systole among all surfaces of same genus. Let γ be a simple closed geodesic. Then γ intersects two distinct systoles (and distinct from γ).

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Received 15 April 2004