1 Introduction

Abstract

Let denote the upper half-plane or the upper half-disk of center and radius . In this paper we classify the solutions to the Neumann problem

where , with the finite energy condition . As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.

 

The geometric Neumann problem

[5mm] for the Liouville equation



José A. Gálvez, Asun Jiménez and Pablo Mira

 

Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain; e-mails: jagalvez@ugr.es, asunjg@ugr.es
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, E-30203 Cartagena (Murcia), Spain; e-mail: pablo.mira@upct.es

1 Introduction

The Liouville equation has a natural geometric Neumann problem attached to it, that comes from the following question:

Let be a domain with smooth boundary . What are the conformal Riemannian metrics on having constant curvature , and constant geodesic curvature along each boundary component of ? Here, we assume that the metric extends smoothly to the boundary .

An important property of the Liouville equation is that it is conformally invariant. Thus, it is not very restrictive to consider only simple symmetric domains , such as disks, half-planes or annuli. The most studied case is when . In that situation, we are led to the Neumann problem

(1)

where the first equation tells that the conformal metric has constant curvature , and the free boundary condition gives that has constant geodesic curvature for that metric. The above problem was fully solved by Zhang [Zha] (in the finite energy case) and Gálvez-Mira [GaMi] (in general), as an extension of previous results in [LiZh, Ou] (see also [ChLi, ChWa, HaWa]).

Recently, there has been some work on the geometric Neumann problem for Liouville’s equation in in the presence of a boundary singularity, i.e. the problem

with and . In [JWZ], Jost, Wang and Zhou gave a complete classification of the solutions to the above problem under the following assumptions:

  1. The metric has finite area in , i.e.

    (2)
  2. The boundary has finite length for the metric , i.e.

    (3)
  3. The metric has a boundary conical singularity at the origin, i.e. there exists for some .

  4. .

With these hypotheses, they showed that any solution to corresponds to the conformal metric associated to the sector of a sphere of radius one limited by two circles that intersect at exactly two points, or to the complement of a closed arc of circle in the sphere, possibly composed with an adequate branched covering of the Riemann sphere . In particular, they provided explicit analytic expressions for all these solutions.

Our main objective in this paper is to provide several improvements of the Jost-Wang-Zhou theorem. These are included in Theorem 2, Theorem 3, Theorem 4 and Corollary 4.

In Theorem 4 we will remove the last three hypotheses of the above list in the Jost-Wang-Zhou result, and prove that any solution to of finite area is a canonical solution. These canonical solutions have explicit analytic expressions and a simple geometric interpretation as the conformal factor associated to basic regions of -dimensional space forms, up to composition with suitable branched coverings of if (see Section 2). For the case we recover the solutions obtained in [JWZ], together with some new solutions corresponding to the case that the boundary singularity at the origin is not conical; we do not prescribe here any asymptotic behavior at the origin, nor the finite length condition (3).

In Theorem 2 we give a general classification of all the solutions to , without any integral finiteness assumptions, in the spirit of [GaMi]. We show that the class of solutions to is extremely large, but still it can be described in terms of entire holomorphic functions satisfying some adequate properties. As a matter of fact, we give such a result not only in but also in an arbitrary half-disk . That is, we also give a general classification result for the solutions to the local problem

In Theorem 3 we classify the solutions to the local problem that satisfy the finite area condition

(4)

and give a general procedure to construct all of them. In particular, we describe the asymptotic behaviour at the origin of any solution to that satisfies (4). This is a generalization to the case of boundary singularities of the well-known results in [Bry, ChWa, Hei, Nit, War] which describe the asymptotic behaviour of metrics of constant curvature and finite area in the punctured disk .

In Corollary 4 we extend Theorem 4 to the case of an arbitrary number of boundary singularities. This solves a problem posed in [JWZ], under milder hypotheses. The basic examples of conformal metrics of constant curvature with boundary singularities and constant geodesic curvature along each boundary component are the ones determined by circular polygons in , but there are many others. To obtain this larger family, we consider immersed circular polygons for which we allow self-intersections, and give a differential-topological criterion (Alexandrov embeddedness) for them to generate such metrics, see Definition 5. With this, Corollary 4 proves the converse: any conformal metric of finite area and constant curvature on (or equivalently on the unit disk ), with finitely many boundary singularities and constant geodesic curvature along each boundary component, is one of those circular polygonal metrics constructed from Alexandrov-embedded, possibly self-intersecting, circular polygons. Analytically, those metrics will not have simple explicit expressions; yet, one can still give some analytic information about them. In Corollary 5 we will describe for the moduli space of these metrics, by parametrizing it in terms of their associated Schwarzian maps, which have simple explicit expressions.

We have organized the paper as follows. In Section 2 we will present the canonical solutions, together with their geometric interpretation and their basic properties. Section 3 contains some preliminaries. In Section 4 we will study the local problem at a boundary singularity, and prove Theorem 2. In Section 5 we will prove Theorem 3, which describes all the solutions to that satisfy the finite energy condition (4). In Section 6 we will prove Theorem 4, which states that any finite area solution to is a canonical solution. In Section 7 we will prove Corollary 4 and Corollary 5 on the classification of conformal metrics of constant curvature with a finite number of boundary singularities.

2 The canonical solutions

Our objective in this section is to describe, both analytically and geometrically, an explicit family of solutions to satisfying the finite energy condition (2). We will prove in Theorem 4 that these are actually all the finite energy solutions to .

In all that follows, we assume that , without loss of generality.

2.1 Analytic description

Definition 1.

A canonical solution is a function of one of the following types:

  1. given by

    (5)

    where and satisfy for all .

  2. given by

    (6)

    where and , satisfy for all . Here, , where .

Let us observe some elementary properties of these canonical solutions, and explain for what choices of the constants and they exist.

The function given by (5) is well defined in if for all , . However, if , is well defined if and only if . In other words, if and only if the distance from the point to the sector is bigger than . A simple analysis shows that this happens:

  • for if and only if , or and with .

  • for if and only if , , and when . Here, and with .

Besides, if the function satisfies the finite area condition

for every .

In the other cases, if it holds for all (and not just for all ), then the metric trivially has finite area. Otherwise, it means that if , or when . But in these cases we clearly have infinite area at the origin.

As a consequence, is a well defined function in with finite area if and only if for all , which is the condition of Definition 1. Observe that when .

Analogously the function given in (6) is well defined in and has finite area if and only if for all .

In particular, if the condition for all is satisfied for every . However, in the other cases, we need to impose that the distance from the point to the strip is bigger that . This condition happens

  • for if and only if or

  • for if and only if or

This analysis together with a simple computation shows that these canonical solutions are indeed finite area solutions to problem .

Lemma 1.

Any canonical solution is a solution to the geometric Neumann problem satisfying

where the constants associated to the problem are given by the following expressions in terms of , and :

  1. For as in (5),

    (7)
  2. For as in (6),

    (8)

2.2 Geometric description

Let denote the -dimensional space form of constant curvature , which will be viewed as where

and is the Riemannian metric on given by

(9)

So, a regular curve in has constant curvature if and only if its image is a piece of a circle in .

Definition 2.

Let be two different circles in such that , and let be any of the regions in which and divide . Assume that is contained in . Then is called a basic domain of .

Let now be a basic domain equipped with the metric in (9). Note that one can conformally parametrize by a biholomorphism such that is a point , and in the case that is not a single point is also some .

It is then clear from this process that the pull-back metric produces a conformal metric of constant curvature in , which has constant geodesic curvature along and , and a singularity at the origin. Also, this metric trivially has finite area, so we have a solution to that satisfies (2).

A similar process can be done if , by considering to be the complement of an arc of a circle in . This would correspond in some sense to taking in the above process.

Furthermore, if and consists of two points, we can easily create other finite area solutions to , starting from the basic region . For that, it suffices to consider a finite-folded branched holomorphic covering of , with branching points at and . If we denote this branched covering by , and consider , the pullback metric of via again describes as before a finite area solution to .

This construction provides a geometric interpretation of the canonical solutions. Indeed, we have:

Fact: Let be a canonical solution. Then is the pullback metric on of either:

  1. some basic region in , or

  2. the complement in of a closed arc of a circle,

possibly composed with a suitable branched covering of in case .

We do not give a direct proof of this fact, since it will become evident from the proof of our main results. See Section 7.

3 Preliminaries

Let us start by explaining the classical relationship between the Liouville equation and complex analysis. From now on we will identify and , and write for points in the domain of a solution to the Liouville equation.

Theorem 1.

Let denote a solution to in a simply connected domain . Then there exists a locally univalent meromorphic function (holomorphic with if ) in such that

(10)

Conversely, if is a locally univalent meromorphic function (holomorphic with if ) in , then (10) is a solution to in .

Up to a dilation, we will assume from now on that . Also, observe that the function in the above theorem, which is called the developing map of the solution, is unique up to a Möbius transformation of the form

(11)
Remark 1.

The developing map has a natural geometric interpretation: if is a solution to , then its developing map provides a local isometry from to , where is given by (9).

There is another relevant holomorphic function attached to any solution of the Liouville equation. We will denote it by , and it is given by the formulas below, where is the developing map of :

(12)

Here, by definition (and ), and is the classical Schwarzian derivative of the meromorphic function with respect to . Observe that is holomorphic, i.e. it does not have poles, and it does not depend on the choice of the developing map . We will call it the Schwarzian map associated to the solution .

The following lemma gives some basic local properties of a solution to the geometric Neumann problem for the Liouville equation along the boundary. It is a consequence of some arguments in [GaMi], but we give a brief proof here for the convenience of the reader.

Lemma 2.

Let , and let be a solution to

Then:

  1. The Schwarzian derivative map of , defined by (12), takes real values along , and extends holomorphically to the whole disk by

  2. The developing map of can be extended to as a locally univalent meromorphic function.

  3. is a regular parametrization of a piece of a circle in .

Proof.

By the Neumann condition along , we have

for every . Thus, holds immediately by Schwarzian reflection.

For , we only need to recall that if is a holomorphic function in a simply connected domain, then the equation always has a locally univalent meromorphic solution , which is unique up to linear fractional transformations. In our case, we have on , and so follows from .

Finally, is clear from the fact that the developing map defines a local isometry from into , and has constant curvature for the metric , by the Neumann condition . ∎

For the proof of Theorem 2, we will also need the following elementary lemma.

Lemma 3.

Let , with , and let be a function such that . Then, there exists a well defined function on the topological annulus such that for all .

Moreover, if is a meromorphic function then so it is .

4 The local problem: proof of Theorem 2.

A general description of all solutions to , in the spirit of the main result in [GaMi], is given by the following theorem. We let denote }.

Theorem 2.

Let be a solution of . Then there exists a meromorphic function such that can be computed from (10) for a locally univalent meromorphic function given by one of the following expressions:

  1. , with and for any ,

  2. , with for any ,

  3. , with and for any .

Here, is a Möbius transformation and is holomorphic with if .

Conversely, let be a locally univalent meromorphic function, holomorphic with if , constructed from a meromorphic function as in above. Then, the function given by (10) is a solution of problem .

Remark 2.

Theorem 2 also provides all the solutions of the global problem . For that, it is enough to consider in the previous theorem, that is, to change by .

Proof.

Let be a solution of problem , and consider an associated developing map . As explained in Lemma 2, the Schwarzian map of , given by (12), extends holomorphically to the punctured disk . Consider now the covering map , from to , which is a local biholomorphism. Then, in the region of such that we can take the meromorphic map given by

(13)

Moreover, the Schwarzian of satisfies

(14)

As is globally defined and holomorphic in , we see by the existence of solutions to the Schwarzian equation that can be extended to a locally univalent meromorphic function globally defined on . In addition, since the right hand side of (14) is -periodic, and since solutions to the Schwarzian equation are unique up to Möbius transformations, we see that the meromorphic function satisfies

(15)

for a certain Möbius transformation .

As explained in Lemma 2, lies on a circle for , and lies on another circle for . We will study the behavior of in terms of the relative position of both circles.

Case 1: intersects in two points or they coincide.

If and share at least two points, then we can consider a Möbius transformation such that is the circle and is the circle given by a straight line passing through the origin and . For that, observe that is the composition of a Möbius transformation which maps the previous two points of into , and a rotation with respect to the origin.

From (13) and (15), the new locally univalent meromorphic maps and satisfy

(16)

and

(17)

for a certain Möbius transformation .

For any real number we have . Hence, by the Schwarz reflection principle,

(18)

Thus, from (17) and (18),

(19)

And, since the set lies on the circle and has no empty interior in , then

But a Möbius transformation is determined by the image of three points, and passes through and . So, if we take an arbitrary point we easily obtain that

Therefore, from (17), we get

where for a real constant . Finally, in order to obtain we observe that the new meromorphic function

(20)

satisfies

So, from Lemma 3, there exists a well defined meromorphic function in the punctured disk such that

Hence, (18) and (20) give

(21)

with and , .

In particular, the developing map of any solution of the local problem when and have at least two common points is given, from (16) and (21), by

(22)

for certain complex constants , with , which determine the Möbius transformation .

Remark 3.

If , then and so .

Case 2: intersects in a unique point.

Let be the common point of the circles and . Then we consider a Möbius transformation that maps to the circle and maps to the circle . For that, observe that can be seen as a Möbius transformation mapping into which maps to , composed with a homothety.

As in the previous case, we define the new locally univalent meromorphic maps and which satisfy (16), (17), (18) and (19).

Since the set lies on the circle and has no empty interior there, then

Therefore, and so, from (17),

Now, the new meromorphic function satisfies for all . Hence, using Lemma 3 for the meromorphic function , there exists a well defined meromorphic function in the punctured disk such that

(23)

Moreover, from (18), , .

With all of this, the developing map of any solution of the local problem when and have only one common point is given, from (16) and (23), by

(24)

for certain complex constants , with , which determine the Möbius transformation .

Case 3: does not intersect .

In this case, it is well known that there exists a Möbius transformation such that the image of the circles and are the circles centered at the origin with radii and , respectively.

We start by considering the locally univalent meromorphic maps and , which satisfy again (16) and (17) for a certain Möbius transformation .

Given a real number we have . So, from the Schwarz reflection principle

(25)

In addition, from (17),

Thus, proceeding as in the previous cases, we have

that is, for any .

Therefore, and so, from (17),

Then we can apply Lemma 3 to the meromorphic function

Hence, there exists a well defined meromorphic function in such that

for the negative real constant . Moreover, from (25), , for any .

As a consequence, the developing map of any solution of the local problem when and have no common points is given, from (16), by

(26)

for certain complex constants , with , which determine the Möbius transformation .

This completes the proof of the first part of Theorem 2. Also, the converse part of the theorem is just a straightforward computation, so we are done. ∎

5 Finite area: proof of Theorem 3

The following result describes the solution to problem under the additional assumption (4) of finite area.

Theorem 3.

Let be a solution of that satisfies the finite energy condition (4). Then, its developing map is given by the cases (i) or (ii) in Theorem 2, and does not have an essential singularity at the origin.

In particular, can be continuously extended to the origin, and the Schwarzian map of has at most a pole of order two at .

Proof.

Let us start by explaining that it suffices to prove the result for the case . Indeed, let be a solution of (4) for a constant and an associated developing map. Now, let us consider the function given by (10) for the map and . Then, is also a solution of , but in this case for , and it also satisfies (4) since

In other words, if the result is true for , it will automatically be true for , as claimed.

Thus, let be the developing map of a solution to for . First, let us prove that the lengths of the semicircles

for the metric of constant curvature , tend to zero when goes to zero.

If we denote by the length of the semicircle for , and write , we have from (10)