The topology of balls and Gromov hyperbolicity of Riemann surfaces

nse ofty.

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Differential Geometry and its Applications 21 (2004) 317–335www.elsevier.com/locate/difgeo

The topology of balls and Gromov hyperbolicityof Riemann surfaces

Ana Portilla1, José M. Rodríguez∗,1, Eva Tourís2

Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad,30, Leganés 28911, Madrid, Spain

Received 15 January 2003; received in revised form 3 October 2003

Available online 24 June 2004

Communicated by O. Kowalski

Abstract

We prove that every ball in any non-exceptional Riemann surface with radius less or equal than12 log3 is either

simply or doubly connected. We use this theorem in order to study the hyperbolicity in the Gromov seRiemann surfaces. The results clarify the role of punctures and funnels of a Riemann surface in its hyperbolici 2004 Elsevier B.V. All rights reserved.

MSC:30F20; 30F45

Keywords:Gromov hyperbolicity; Riemann surface; Funnel; Puncture

1. Introduction

A good way to understand the important connections between graphs and Potential TheRiemannian manifolds (see, e.g.,[2,5,9,14,16–18,22,23,28]) is to study the Gromov hyperbolic spaceThis approach allows to establish a general setting to work simultaneously with graphs and manifthe context of metric spaces. Besides, the idea of Gromov hyperbolicity grasps the essence of ne

* Corresponding author.E-mail addresses:[email protected] (A. Portilla), [email protected] (J.M. Rodríguez), [email protected]

(E. Tourís).1 Research partially supported by a grant from DGI (BFM 2000-0022), Spain.2 Research supported by a grant from DGI (BFM 2000-0022), Spain.

0926-2245/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.difgeo.2004.05.006

http://www.elsevier.com/locate/difgeo

318 A. Portilla et al. / Differential Geometry and its Applications 21 (2004) 317–335

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curved spaces, and has been successfully used in the theory of groups (see, e.g.,[10] and the referencetherein).

Although there exist some interesting examples of hyperbolic spaces (see the examples afteDefini-tion 2.1), the literature gives no good guide about how to determine whether or not a space is hypThis limitation can be somehow got round, since the theory allows to obtain powerful results abohyperbolic spaces which have hyperbolic universal coverings. As topological “obstacles” may prspace from being hyperbolic, the possibility of studying its universal covering instead, which is afree of obstacles, implies a substantial simplification, and sometimes let us extract important inforabout the space itself (see[19]).

However, as was stated above, the characterization of hyperbolic spaces remains open. Receninteresting results about the hyperbolicity of Euclidean bounded domains with their quasihypmetric have made significant progress in this direction (see[3] and the references therein).

Originally, we were interested in studying when non-exceptional Riemann surfaces equippedPoincaré metric were Gromov hyperbolic. However, we have proved two theorems on hyperbfor general metric spaces, which are interesting by themselves (seeSection 2) and have importanconsequences for Riemann surfaces (seeSection 3). Although one should expect Gromov hyperbolicitynon-exceptional Riemann surfaces due to its constant curvature−1, this turns out to be untrue in genersince topological obstacles can impede it: for instance, the two-dimensional jungle-gym (aZ2-coveringof a torus with genus two) is not hyperbolic. Let us recall that in the case of modulated plane doquasihyperbolic metric and Poincaré metric are equivalent.

We prove in[26] that there is no inclusion relationship between hyperbolic Riemann surfaces ausual classes of Riemann surfaces, such asOG, OHP , OHB , OHD, surfaces with hyperbolic isoperimetrinequality, or the complements of these classes (even in the case of plane domains). This facthe study of hyperbolic Riemann surfaces more complicated and interesting. One can find reshyperbolicity of Riemann surfaces in[24–26].

Here we present the outline of the main results. We refer to the next sections for the definitiothe precise statements of the theorems.

In Section 2we obtain some lower bounds on the hyperbolicity constants of metric spaces, whicbe useful inSection 3. In Section 3we study the role of punctures and funnels of a Riemann surfaits hyperbolicity.

The main aim in this paper is obtaining global results on hyperbolicity from local information.was the idea that led us to identify the punctures and funnels of a surfaceS∗ with closed sets{En}n

removed from an original surfaceS, in such a way thatS∗ = S \ ⋃n En.

Theorem 3.2allows, in many cases, to forget punctures and funnels in order to study the hyperbof a Riemann surface; this fact is a significant simplification in the topology of the surface, and themakes easier the problem. Besides, we have determined which are the relevant parametehyperbolicity constant ofS∗. If we consider just punctures,Theorem 3.4gives a result with a statememuch simpler thanTheorem 3.2.

In order to proveTheorem 3.4we need a universal result on the topology of balls in Riemann sur(seeTheorem 3.1), which is interesting by itself: it says that every ball in any non-exceptional Riemsurface with radius less or equal than1

2 log 3 is either simply or doubly connected.Theorem 3.1is aprecise answer in our context to the question: when do geometric constraints imply topologicaThis is an attractive topic of research, as plenty of publications in first-rate quality journals showe.g.,[7,11–13]).

A. Portilla et al. / Differential Geometry and its Applications 21 (2004) 317–335 319

the

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nds justee, e.g.,

,

these

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sincenction.

unded

As a consequence ofTheorem 3.4, we have obtained interesting examples of stability ofhyperbolicity of Riemann surfaces (seeCorollary 3.3).

We also prove a general criteria which guarantees that many surfaces are not hyperbolic (seTheo-rem 3.3).

It is a remarkable fact that almost every constant appearing in the results of this paper depeon a small number of parameters. This is a common place in the theory of hyperbolic spaces (sTheorems A, B and C, andLemma B) and is also typical of surfaces with curvature−1 (see, e.g., theCollar Lemma in[21] and[27], andTheorem 3.1).

Notations. We denote byX or Xn geodesic metric spaces. BydX, LX and BX we shall denoterespectively, the distance, the length and the balls in the metric ofX.

We denote byS or Si non-exceptional Riemann surfaces. We assume that the metric defined onsurfaces is the Poincaré metric.

Finally, we denote byki positive constants which can assume different values in different theore

2. Results in metric spaces

In our study of hyperbolic Gromov spaces we use the notations of[10]. We give now the basic factabout these spaces. We refer to[10] for more background and further results.

Definition 2.1. Let us fix a pointw in a metric space(X,d). We define theGromov productof x, y ∈ X

with respect to the pointw as

(x|y)w := 1

2

(d(x,w) + d(y,w) − d(x, y)

)� 0.

We say that the metric space(X,d) is δ-hyperbolic(δ � 0) if

(x|z)w � min{(x|y)w, (y|z)w

} − δ,

for every x, y, z,w ∈ X. We say thatX is hyperbolic (in the Gromov sense) if the value ofδ is notimportant.

It is convenient to remark that this definition of hyperbolicity is not universally accepted,sometimes the word hyperbolic refers to negative curvature or to the existence of Green’s fuHowever, in this paper we only use the wordhyperbolicin the sense ofDefinition 2.1.

Examples.

(1) Every bounded metric spaceX is (diamX)-hyperbolic (see, e.g.,[10, p. 29]).(2) Every complete simply connected Riemannian manifold with sectional curvature which is bo

from above by−k, with k > 0, is hyperbolic (see, e.g.,[10, p. 52]).(3) Every tree with edges of arbitrary length is 0-hyperbolic (see, e.g.,[10, p. 29]).


h

gcs, thisnected.

s

.g.,

o

-

Definition 2.2. If γ : [a, b] → X is a continuous curve in a metric space(X,d), we can define the lengtof γ as

L(γ ) := sup

{n∑

i=1

d(γ (ti−1), γ (ti)

): a = t0 < t1 < · · · < tn = b

}.

We say thatγ is a geodesicif it is an isometry, i.e.,L(γ |[t,s]) = d(γ (t), γ (s)) = |t − s| for everys, t ∈ [a, b]. We say thatX is ageodesic metric spaceif for everyx, y ∈ X there exists a geodesic joininx andy; we denote by[x, y] any of such geodesics (since we do not require uniqueness of geodesinotation is ambiguous, but it is convenient). It is clear that every geodesic metric space is path-con

Definition 2.3. If X is a geodesic metric space andJ = {J1, J2, . . . , Jn}, with Jj ⊆ X, we say thatJ is δ-thin if for every x ∈ Ji we have thatd(x,

⋃j �=i Jj ) � δ. If x1, x2, x3 ∈ X, a geodesic triangle

T = {x1, x2, x3} is the union of three geodesics[x1, x2], [x2, x3] and[x3, x1]. The spaceX is δ-thin (orsatisfies theRips conditionwith constantδ) if every geodesic triangle inX is δ-thin.

A basic result is that hyperbolicity is equivalent to Rips condition:

Theorem A [10, p. 41]. Let us consider a geodesic metric spaceX.

(1) If X is δ-hyperbolic, then it is4δ-thin.(2) If X is δ-thin, then it is4δ-hyperbolic.

We present now the class of maps which play the main role in the theory.

Definition 2.4. A function between two metric spacesf :X → Y is aquasiisometryif there are constanta � 1, b � 0 with

1

adX(x1, x2) − b � dY

(f (x1), f (x2)

)� adX(x1, x2) + b, for everyx1, x2 ∈ X.

A such function is called an(a, b)-quasiisometry. An (a, b)-quasigeodesicin X is an (a, b)-quasiisometry between an interval ofR and X. An (a, b)-quasigeodesic segmentin X is an (a, b)-quasiisometry between a compact interval ofR andX.

Quasiisometries are important since they are the maps which preserve hyperbolicity (see, e[10,p. 88]). Notice that a quasiisometry can be discontinuous.

Along this paper we will work with topological subspaces of a geodesic metric spaceX. There is anatural way to define a distance in these spaces:

Definition 2.5. If X0 is a path-connected subset of a geodesic metric space(X,d), then we associate tit the restricted distance

dX0(x, y) := dX|X0(x, y) := inf{L(γ ): γ ⊂ X0 is a continuous curve joiningx andy

}� dX(x, y).

If X0 is not path-connected, we also use this definition ifx and y belong to the same pathconnected component ofX0; if x andy belong to distinct path-connected components ofX0, we definedX0(x, y) := ∞.


rbolic

e the

n

c

.

The following result will be useful in order to decide that a geodesic metric space is not hype(seeTheorem 3.3).

Theorem 2.1. Let us consider a geodesic metric spaceX, and X1,X2 ⊂ X two geodesic metricspaces such thatX1 ∩ X2 = η1 ∪ η2, with ηi compact sets,diamXi

(ηj ) � c1 for any i, j = 1,2,and dX(η1, η2) � c2. Then there exists a geodesic triangleT = {a, b, c} in X and x ∈ [a, b] withdX(x, [a, c] ∪ [b, c]) � c2/2− c1.

Remark. We will see in the proof of the theorem that the conclusion is also true if we changhypothesis “ηi are compact sets”, by “there exist geodesicsγi in Xi joining η1 andη2, with LX(γi) =dXi

(η1, η2)”.

Proof. Without loss of generality, we can assume thatc2 � 2c1, since if this was not so, the conclusiois clear. Sinceη1, η2 are compact sets, we have that there exist geodesicsγi in Xi joining η1 andη2, withLX(γi) = dXi

(η1, η2).Without loss of generality, we can assume thatLX(γ1) � LX(γ2); then it is not difficult to see thatγ1 is

also a geodesic inX: It is clear that a geodesicγ in X such thatLX(γ ) = dX(η1, η2) must be completelycontained inX1 or in X2. If γ1 = [a, b], with a ∈ η1, b ∈ η2, let us consider a geodesicγ ′

2 in X2 joining a

andb. Let us callc to the middle point ofγ ′2. We consider geodesics[a, c], [b, c] in X, and the geodesi

triangleT in X with these three geodesics joininga, b, c.We see now that[a, c] cannot contain a geodesic connectingη1 with η2 in X1: If [a, c] contains such

geodesic, we call itg; thenLX(g) � dX(η1, η2) � c2. If LX(γ ′2) = 2r , then we have that

dX(c, η1 ∪ η2) = dX2(c, η1 ∪ η2)

� min{dX2(c, a) − diamX2(η1), dX2(c, b) − diamX2(η2)

}� r − c1.

Consequentlyr = LX(γ ′2)/2 = LX([a, c]) � dX(c, η1 ∪ η2) + LX(g) � r − c1 + c2, which is a

contradiction withc2 � 2c1. Hence,LX([a, c] ∩ X1) � c1 anddX(p,η1) � c1 for everyp ∈ [a, c] ∩ X1.A similar result holds for[b, c].

Consequently, ifx is the middle point ofγ1, then

dX

(x, [a, c] ∪ [b, c]) � dX(η1, η2)/2− c1 � c2/2− c1. �

In the applications we usually knowdX2(η1, η2), but we do not have any lower bound ofdX(η1, η2)

at all. We can obtain a similar result toTheorem 2.1with just a bound ofdX2(η1, η2), if we work withquasigeodesic triangles.

Definition 2.6. Let us consider three quasigeodesicsJ1 starting inx1 and finishing inx2, J2 starting inx2

and finishing inx3, J3 starting inx3 and finishing inx1, in a metric space. We say thatT = {J1, J2, J3} isan(a, b)-quasigeodesic triangleif J1, J2, J3 are(a, b)-quasigeodesics.

We need the following elementary result.

Lemma A [20, Lemma 3]. Let us consider an(a, b)-quasigeodesicq1 : [α,β] → X and two continuouscurves with arc-length parametrizationq0 : [α − d1, α] → X, q2 : [β,β + d2] → X, verifying q0(α) =q1(α) andq2(β) = q1(β). Then the curveq := q0∪q1∪q2 is an(a, b+(1+a−1)(d1+d2))-quasigeodesic


s

hat

e

at

at

.

owing

The next result will be especially useful to decide that some spaces are not hyperbolic (seeCorol-lary 2.1, Theorems 3.2 and 3.3, andLemma 3.1).

Theorem 2.2. Let us consider a geodesic metric spaceX, and X1,X2 ⊂ X two geodesic metricspaces such thatX1 ∩ X2 = η1 ∪ η2, with ηi compact sets,dX2(η1, η2) � c2 and diamXi

(ηj ) � c1 fori, j = 1,2. Then there exists a(1,2c1)-quasigeodesic triangleT = {A,B,C} in X and x ∈ A withdX(x,B ∪ C) � c2/4.

Remark. The conclusion ofTheorem 2.2also holds ifη1 intersectsη2 (and even ifη1 = η2); in this casewe consider thatη1 andη2 are disjoint sets inX2 (they are identified if we pasteX1 andX2 in order toobtainX).

Theorem 2.2is a direct consequence of the following result.

Theorem 2.2′. Let us consider a geodesic metric spaceX, andX1,X2 ⊂ X two geodesic metric spacesuch thatX1 ∩ X2 = η1 ∪ η2, with ηi compact sets,dX1(η1, η2) � dX2(η1, η2), dX2(η1, η2) � c2 anddiamX1(ηj ) � c1 for j = 1,2. Then there exists a(1,2c1)-quasigeodesic triangleT = {A,B,C} in X

andx ∈ A with dX(x,B ∪ C) � c2/4.

Proof. Sinceη1, η2 are compact sets, we have that there exist geodesicsγi in Xi joining η1 and η2,with LX(γ1) = dX1(η1, η2) � LX(γ2) = dX2(η1, η2). Without loss of generality, we can assume tLX(γ2) = c2.

Let us denote bya ∈ η1 andb ∈ η2 the end points ofγ2, and byc its middle point. We have that thtwo subcurves ofγ2 joining a with c, andb with c (both of lengthc2/2), are geodesics inX: If there issome curveg in X joining a andc with LX(g) < c2/2, then there is some curveg0 ⊆ g joining c with η1

or η2 in X2 with LX(g0) < c2/2; consequently, we can construct a curve joiningη1 andη2 in X2 shorterthanγ2. If there is some curveg in X joining b with c with LX(g) < c2/2, we have the same result.

Let us consider the triangleT in X with sides[a, c], [b, c] ⊂ γ2 andγ3, whereγ3 is a continuous curvejoining a with b in X1 in the following way:γ3 is the union ofγ1 and two geodesics inX1 joining a withthe end point ofγ1 belonging toη1, andb with the end point ofγ1 belonging toη2. By Lemma A wehave thatγ3 is a (1,2c1)-quasigeodesic, since diamX1(ηj ) � c1 for j = 1,2. We definex as the middlepoint of [a, c].

We only need to prove thatdX(x, γ3) = dX(x, [b, c]) = c2/4:Let us denote byp a point inγ3 such thatdX(x, γ3) = dX(x,p). Seeking a contradiction, suppose th

there is some curveh in X joining x andp with LX(h) < c2/4. Then there is some curveh0 ⊆ h joiningx with η1 or η2 in X2 with LX(h0) < c2/4; consequently, we can construct a curve joiningη1 andη2 inX2 shorter thanγ2. ThereforedX(x, γ3) � c2/4; sincedX(x, a) = c2/4, we have thatdX(x, γ3) = c2/4.

Let us denote byq a point such thatdX(x, [b, c]) = dX(x, q). Seeking a contradiction, suppose ththere is some curver in X joining x andp with LX(r) < c2/4. If r intersectsη1 ∪ η2, we can use thesame argument as in the previous case. If this was not so, the curver is contained inX2; sinceγ2 is ageodesic inX2, we obtainLX(r) = dX(x,p) = dX(x, [c, b]) = dX(x, c) = c2/4, which is a contradictionThereforedX(x, [c, b]) � c2/4; sincedX(x, c) = c2/4, we have thatdX(x, [c, b]) = c2/4. �

In order to useTheorem 2.2to guarantee that some spaces are not hyperbolic, we need the follelementary result.


dges

lts of

ic

re

tarrow”parts

y

ee thed.

Lemma B [20, Lemma 4]. For eachδ, b � 0 anda � 1, there exists a constantK = K(δ, a, b) with thefollowing property:

If X is a δ-hyperbolic geodesic metric space andT ⊆ X is an(a, b)-quasigeodesic triangle, thenT isK-thin.

Corollary 2.1. Let us consider a graphG which is a geodesic metric space, with a sequence of e{en}n such that the graphG \ en is a geodesic metric space for everyn, andlimn→∞ L(en) = ∞. ThenG

is not hyperbolic.

We finish this section with two theorems which will be very useful in the proof of the main resuthis paper. In order to state them, we need a definition.

Definition 2.7. We say that a geodesic metric spaceX has adecomposition, if there exists a familyof geodesic metric spaces{Xn}n∈Λ with X = ⋃

n∈Λ Xn and Xn ∩ Xm = ⋃i∈Inm

ηinm, where for each

n ∈ Λ, {ηinm}m,i are pairwise disjoint closed subsets ofXn (ηi

nm = ∅ is allowed); furthermore any geodessegment inX meets at most a finite number ofηi

nm’s.We say thatXn, with n ∈ Λ, is a(k1, k2, k3)-tree-pieceif it satisfies the following properties:

(a) �Inm � 1 (then we can writeηinm = ηnm), X \ ηnm is not connected form �= n if �Inm = 1, anda, b

are in different components ofX \ ηnm for anya ∈ Xn \ ηnm, b ∈ Xm \ ηnm.(b) diamXn

(ηnm) � k1 for everym �= n, and there existsAn ⊆ Λ, such that diamXn(ηnm) � k2 dXn

(ηnm,

ηnk) if m �= k andm,k ∈ An, and∑

m/∈AndiamXn

(ηnm) � k3.

We say that a geodesic metric spaceX has atree-decompositionif it has a decomposition and theexist positive constantsk1, k2, k3, such that everyXn, with n ∈ Λ, is a(k1, k2, k3)-tree-piece.

We wish to emphasize that condition diamXn(ηnm) � k1 is not very restrictive: if the space is “wide” a

every point (in the sense of long injectivity radius, as in the case of simply connected spaces) or “nat every point (as in the case of trees), it is easier to study its hyperbolicity; if we can found narrow(asηnm) and wide parts, the problem is more difficult and interesting.

Remarks.

(1) Obviously, condition (b) is required only forηnm, ηnk �= ∅.(2) The setsΛ andAn do not need to be countable.(3) Condition (a) for everyn ∈ Λ guarantees that the graphR = (V ,E) constructed in the following wa

is a tree:V = ⋃n∈Λ{vn} and[vn, vm] ∈ E if and only if ηnm �= ∅.

(4) If X is a Riemann surface and{Xn}n∈Λ are bordered Riemann surfaces andηnm ⊂ ∂Xn ∩ ∂Xm,condition “a, b are in different components ofX \ ηnm for anya ∈ Xn \ ηnm, b ∈ Xm \ ηnm” in (a), isa consequence of “X \ ηnm is not connected”.

The following result can be applied to the study of the hyperbolicity of Riemann surfaces (sproof of Propositions 3.1 and 3.2). In [20] explicit expressions for the constants involved are supplie


d inof

é metric.pens

s”,

Theorem B [20, Theorem 1]. Let us consider a tree-decomposition{Xn}n∈Λ of a geodesic metricspaceX. ThenX is δ-hyperbolic if and only if there exists a constantk4 such thatXn is k4-hyperbolicfor everyn ∈ Λ. Furthermore, ifX is δ-hyperbolic, thenk4 only depends onδ, k1, k2 and k3; if thereexistsk4, thenδ only depends onk1, k2, k3 andk4.

Definition 2.8. We say that two geodesic metric spacesX and Y (in this order) havecomparabledecompositions, if there exist decompositions{Xn}n∈Λ of X and {Yn}n∈Λ of Y , and constantski , withthe following properties:

(a) If Xn ∩ Xm = ⋃i∈Inm

ηinm, thenYn ∩ Ym = ⋃

i∈Inmσ i

nm, andσ inm = ∅ if and only if ηi

nm = ∅.(b) For anyn,m, i, diamXn

(ηinm) � k1 and diamYn

(σ inm) � k1.

(c) We can splitΛ into F ∪ G andF into F1 ∪ F2 with:(c1) If n ∈ G, Xn is a(k1, k2, k3)-tree-piece.(c2) If n ∈ F , diamXn

(ηinm) � k2 dXn

(ηinm, η

j

nk) and diamYn(σ i

nm) � k2 dYn(σ i

nm, σj

nk) if (m, i) �= (k, j).(c3) If n ∈ F1, for eachηi

nm �= ηj

nk , there exists a geodesicγ ij

mnk in Xn, joining ηinm with η

j

nk , anda (k4, b

ij

mnk)-quasiisometryf ij

mnk :γ ij

mnk → hij

mnk ⊆ Yn, with hij

mnk starting inσ inm and finishing in

σj

nk, and∑

n∈F1

∑m,k,i,j b

ij

mnk � k5, such that for anyx, y ∈ ⋃m,k,i,j γ

ij

mnk , with corresponding

pointsx′, y′ ∈ ⋃m,k,i,j h

ij

mnk , we havek−14 dXn

(x, y) − k5 � dYn(x′, y′).

(c4) If n ∈ F2, there exists a(k4,0)-quasiisometryfn :Xn → Yn, with fn(ηinm) ⊆ σ i

nm.

Remark. The hypothesis diamXn(ηnm) � k2 dXn

(ηnm, ηnk) holds if we havedXn(ηnm, ηnk) � k′

2, sincediamXn

(ηnm) � k1.

The conditions thatXn must verify whenn belongs toF1,F2 or G in Definition 2.8, is not arbitraryat all. In fact, what lies behind is an appropriate modelization for the situation which we will finthe proof ofTheorem 3.2. The following theorem will be one of the important tools in the proofTheorem 3.2. In [20] explicit expressions for the constants involved are supplied.

Theorem C [20, Theorem 2]. Let us assume that two geodesic metric spacesX andY have comparabledecompositions. IfY is δ′-hyperbolic and there exists a constantk6 such thatXn is k6-hyperbolic foreveryn ∈ Λ \ F2, thenX is δ-hyperbolic, withδ a constant which only depends onδ′ andki .

3. Results in Riemann surfaces

In this section we always work with the Poincaré metric; consequently, curvature is always−1. Infact, many concepts appearing here (as punctures or funnels) only make sense with the Poincar

The intuition would say that negative curvature must imply hyperbolicity; in fact this is what hapwhen there are no topological “obstacles” (as in the case of the Poincaré diskD) or if there is a finitenumber of them (see[24, Proposition 3.2]). However, if there are infinitely many topological “obstaclethe hyperbolicity can fail, as in the case of the two-dimensional jungle gym (aZ2-covering of a torus withgenus two).


iemannerbolic

ed to

dary)

it disk

rfaceithh arelicity of

t

anating

exists

r aclosedodesic

anating

cenn

The results in this section are useful since they not only provide many examples of hyperbolic Rsurfaces, but also allow to establish criteria in order to decide whether a Riemann surface is hypor not.

Below we collect some definitions concerning to Riemann surfaces which will be referrafterwards.

An open non-exceptionalRiemann surface (or a non-exceptional Riemann surface without bounS is a Riemann surface whose universal covering space is the unit diskD = {z ∈ C: |z| < 1}, endowedwith its Poincaré metric, i.e., the metric obtained by projecting the Poincaré metric of the unds = 2|dz|/(1 − |z|2) or, equivalently, the upper half planeU = {z ∈ C: Im z > 0}, with the metricds = |dz|/ Im z. Observe that, with this definition, every compact non-exceptional Riemann suwithout boundary is open. With this metric,S is a geodesically complete Riemannian manifold wconstant curvature−1, and thereforeS is a geodesic metric space. The only Riemann surfaces whicleft out are the sphere, the plane, the punctured plane and the tori. It is easy to study the hyperbothese particular cases.

It is well known (see, e.g.,[15, p. 227]) that

(3.1)dD(0, z) = log1+ |z|1− |z| = 2Argtanh|z|, sinh2 dU(z,w)

2= |z − w|2

4 Imz Imw.

Let S be an open non-exceptional Riemann surface with a punctureq (if S ⊂ C, every isolated poinin ∂S is a puncture). Acollar in S aboutq is a doubly connected domain inS “bounded” both byq anda Jordan curve (called the boundary curve of the collar) orthogonal to the pencil of geodesics emfrom q.

A collar in S aboutq of areaα will be called anα-collar and it will be denoted byCS(q,α). A theoremof Shimizu[27] gives that for every puncture in any open non-exceptional Riemann surface, thereanα-collar for every 0< α � 2 (see also[4, Chapter 4.4]).

We say that a curve ishomotopic to a punctureq if it is freely homotopic to∂CS(q,α) for some (andthen for every) 0< α < 2.

We have used the wordgeodesicin the sense ofDefinition 2.2, that is to say, as a global geodesic ominimizing geodesic; however, we need now to deal with a special type of local geodesics: simplegeodesics, which obviously cannot be minimizing geodesics. We will continue using the word gewith the meaning ofDefinition 2.2, unless we are dealing with closed geodesics.

A collar in S about a simple closed geodesicγ is a doubly connected domain inS “bounded” by twoJordan curves (called the boundary curves of the collar) orthogonal to the pencil of geodesics emfrom γ ; such collar is equal to{p ∈ S: dS(p, γ ) < d}, for some positive constantd. The constantd iscalled thewidth of the collar. The Collar Lemma[21] says that there exists a collar ofγ of width d, forevery 0< d � d0, where coshd0 = coth(LS(γ )/2) (see also[4, Chapter 4]).

We say thatS is abordered non-exceptional Riemann surface(or a non-exceptional Riemann surfawith boundary) if it can be obtained deleting an open setV of an open non-exceptional RiemasurfaceR, such that:

(1) S is connected anddS := dR|S (recallDefinition 2.5),(2) any ball inR intersects at most a finite number of connected components ofV ,(3) the boundary ofS is locally Lipschitz.


nnian

whosealr

ere are

spherepositive

esces.

h is

anwot

d

of

d

Any such surfaceS is a bordered orientable Riemannian manifold of dimension 2 and its Riemametric has constant negative curvature−1. It is not difficult to see thatS is a geodesic metric space.

A funnelis a bordered non-exceptional Riemann surface which is topologically a cylinder andboundary is a simple closed geodesic. Given a positive numbera, there is a unique (up to conformmapping) funnel such that its boundary curve has lengtha. Every funnel is conformally equivalent, fosomeβ > 1, to the subset{z ∈ C: 1 � |z| < β} of the annulus{z ∈ C: 1/β < |z| < β}.

Every doubly connected end of an open non-exceptional Riemann surface is a puncture (if thhomotopically non-trivial curves with arbitrary small length) or a funnel (if this was not so).

A Y -pieceis a bordered non-exceptional Riemann surface which is conformally equivalent to awithout three open disks and whose boundary curves are simple closed geodesics. Given threenumbersa, b, c, there is a unique (up to conformal mapping)Y -piece such that their boundary curvhave lengthsa, b, c (see, e.g.,[4, p. 109]). They are a standard tool for constructing Riemann surfaA clear description of theseY -pieces and their use is given in[6, Chapter X.3]and[4, Chapter 3].

A generalized Y-pieceis a non-exceptional Riemann surface (with or without boundary) whicconformally equivalent to a sphere withoutn open disks andm points, with integersn,m � 0 suchthat n + m = 3, the n boundary curves are simple closed geodesics and them deleted points arepunctures. Observe that a generalizedY -piece is topologically the union of aY -piece andm cylinders,with 0� m � 3.

By thecollar of a puncture we mean the 2-collar. Bythecollar of a simple closed geodesic we methe collar of widthd0, where coshd0 = coth(LS(γ )/2). We have that two collars (corresponding to tdistinct punctures, two disjoint geodesics or to one puncture and one geodesic) inS are always disjoin(see, e.g.,[4, p. 112]).

Although the following result is an important tool in the proof ofTheorem 3.4, it is interesting by itselfas well. Let us observe that it gives universal constants which depend neither on the surfaceS nor on thepoint p ∈ S, in a similar way to the Collar Lemma.

Theorem 3.1. Let us consider an open non-exceptional Riemann surfaceS and p ∈ S. If in BS(p, r)

there is a closed curve freely homotopic to a puncture or to a simple closed geodesicγ and r � 12 log 3,

thenBS(p, r) is contained in the collar ofγ . Consequently,BS(p, r) is simply or doubly connected, an∂BS(p, r) has at most two connected components.

Proof. We consider the ballBS(p, r) containing a closed curve freely homotopic to a punctureγ , withr � 1

2 log 3. We also consider a universal covering mapπ : U → S. We can assume, without lossgenerality, thatπ({0 � Rez < 1, Im z > 1/2}) is the 2-collar ofγ , and thatπ(it) = π(1+ it) = p, forsomet > 0. There is a geodesic (except in the pointp) γ1 freely homotopic toγ , starting and finishingin p, with length less than 2r . We consider the liftγ2 of γ1 to U starting init and finishing in 1+ it .By (3.1), we have that

sinh2 r > sinh2 LS(γ1)

2= sinh2 dU(it,1+ it)

2= 1

4t2, t >

1

2sinhr.

Sincer � 12 log 3, we obtainte−r > e−r /(2sinhr) = 1/(e2r − 1) � 1/2. Then (see, e.g.,[15, p. 227]), we

haveBU(it, r) = {(Rez)2 + (Im z − t coshr)2 � t2 sinh2 r} ⊂ {Im z � te−r} ⊂ {Im z > 1/2}.Consequently,BS(p, r) ⊂ CS(q,2), and this fact implies thatBS(p, r) is doubly connected an

∂BS(p, r) is the union of two simple closed curves.


sed

d

eans of

We consider now the ballBS(p, r) containing a closed curve freely homotopic to a simple clogeodesicγ with lengthLS(γ ) = 2l. We consider a universal covering mapπ : U → S with π({Rez =0}) = γ . Thenπ({ρ eiφ : 1 � ρ < e2l , |φ − π/2| < arcsec(coshd)}) is the collar ofγ of width d � d0,if coshd0 = cothl (by the Collar Lemma). Without loss of generality, we can assume thatπ(ie−iθ ) =π(ie2l−iθ ) = p, for some 0< θ < π/2. There is a geodesic (except in the pointp) γ3, freely homotopicto γ , starting and finishing inp, with 2l � LS(γ3) < 2r . We consider the liftγ4 of γ3 to U starting inie−iθ and finishing inie2l−iθ . By (3.1), we have that

sinh2 r > sinh2 LS(γ3)

2= sinh2 dU(ie−iθ , ie2l−iθ )

2, sinhr >

e2l − 1

2el cosθ= sinhl secθ.

If we defines := dS(p, γ ) = dU(i, ie−iθ ), then coshs = secθ and sinhr > sinhl coshs.We will prove nows + r < d0; consequentlyBS(p, r) is contained in the collar ofγ of width d0,

and this fact implies thatBS(p, r) is doubly connected and∂BS(p, r) is the union of two simple closecurves; this will finish the proof ofTheorem 3.1. Observe that the functionf (l) := (2 + coshl)/(2 −coshl) is an increasing function inl ∈ [0, 1

2 log 3], since coshl � cosh(12 log3) = 1/

√3 < 2; then

f (l) � f (0) = 3� e2r for l < r � 12 log3. Therefore, sincel < r � 1

2 log 3, we have

e2r � 2+ coshl

2− coshl= (2+ coshl)(1+ coshl)

(2− coshl)(1+ coshl)= 2+ 3coshl + cosh2 l

2+ coshl − cosh2 l,

and thene2r(sinh2 l − coshl − 1) + 2+ 3coshl + cosh2 l � 0. Consequently,

e2r sinh2 r − e2r sinh2 l � cosh2 l + 2coshl − (e2r − 1

)coshl + 1+ e2r sinh2 r − (

e2r − 1),

e2r sinh2 r − e2r sinh2 l � cosh2 l + 2(1− er sinhr

)coshl + 1+ e2r sinh2 r − 2er sinhr,

e2r(sinh2 r − sinh2 l

)�

(coshl + 1− er sinhr

)2,

er√

sinh2 r − sinh2 l � coshl + 1− er sinhr,

since coshl + 1 > er sinhr (in fact, r � 12 log 3 givese2r − 1 � 2 < 2coshl + 2). Then we have

er � coshl + 1

sinhr +√

sinh2 r − sinh2 l,

r � logcoshl + 1

sinhr +√

sinh2 r − sinh2 l= log

coshl/sinhl +√

cosh2 l/sinh2 l − 1

sinhr/sinhl +√

sinh2 r/sinh2 l − 1

= Argcoshcoshl

sinhl− Argcosh

sinhr

sinhl= d0 − Argcosh

sinhr

sinhl.

Consequently, since sinhr > sinhl coshs, we obtain

s + r � d0 + s − Argcoshsinhr

sinhl< d0.

Hence,BS(p, r) is contained in the collar ofγ . �The hyperbolicity constants of some simple Riemann surfaces can be uniformly bounded by m

the two following propositions. These propositions play a fundamental role in the proof ofTheorem 3.2.


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eterd

s

Proposition 3.1. LetS be a simply or doubly connected bordered non-exceptional Riemann surfacethat LS(∂S) � a. ThenS is δ-hyperbolic, whereδ is a constant which only depends ona.

Remark. As usual, we see a puncture as a geodesic of zero length.

Proof. It is well known thatS is isometric to a bordered surfaceS1 contained inR, whereR is the unitdisk D, the punctured diskD∗ or some annulusNε := {z ∈ C: ε < |z| < 1}, for 0< ε < 1; thenR is theunion ofS1 and at most two other bordered surfaces. Without loss of generality, we can assumeS1 = S.Observe that the diameter inR of each connected component of∂S is less or equal thana.

If R = D or R = D∗, Theorem B(with An = ∅) gives thatS is k4-hyperbolic, sinceD and D∗ arehyperbolic (see[24, Theorem 3.3]), wherek4 is a constant which only depends ona (this is the case ifSis simply connected).

If R = Nε and γ is the simple closed geodesic inNε, we have thatLNε(γ ) � LS(∂S) � a.

Proposition 3.1 in[24] gives thatNε is k5-hyperbolic, wherek5 is a constant which only depends ona.By Theorem B(with An = ∅), S is k′

4-hyperbolic, sinceNε is k5-hyperbolic, wherek′4 is a constant which

only depends onk5 anda.The proof finishes takingδ := max{k4, k

′4}. �

We also need the following result.

Theorem D [26, Theorem 3.6]. Let us consider a non-exceptional Riemann surfaceS (with or withoutboundary) without genus. If there is a decomposition ofS in a union of funnels{Fm}m∈M and generalizedY -pieces{Yn}n∈N with LS(γ ) � a for at least two simple closed geodesicγ ⊂ ∂Yn for everyn ∈ N , thenS is δ-hyperbolic, whereδ is a constant which only depends ona.

We can obtain a similar result toProposition 3.1for triply connected surfaces, usingTheorem D.

Proposition 3.2. Let S be a triply connected bordered non-exceptional Riemann surface, such th∂S

is the union of two simple closed curves verifyingLS(∂S) � a. ThenS is δ-hyperbolic, whereδ is aconstant which only depends ona.

Proof. It is well known thatS is isometric to a bordered surfaceS1 contained in an open non-exceptionRiemann surfaceR, whereR is the unit disk, the punctured disk, an annulus or the union of a generaY -pieceY0 and at most 3 funnels. Without loss of generality, we can assumeS1 = S.

If R is the unit disk, the punctured disk or an annulus, we proceed as in the proof ofProposition 3.1.If this was not so,R is the union ofS and two bordered surfaces. Let us observe that the diamin R of each connected component of∂S is less or equal thana. If g1 and g2 are the simple closecurves in∂S, we denote byγi the simple closed geodesic inR freely homotopic togi (i = 1,2). AsLR(γi) � LS(gi) � a, Theorem Dguarantees thatR is k-hyperbolic, wherek is a constant which onlydepends ona. By Theorem B(with An = ∅), S is δ-hyperbolic, whereδ is a constant which only dependon a. �

The following result will be an important tool in order to prove our next theorems.


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ts

is

dy theof the

Lemma C [1, Lemma 3.1]. Let us consider an open non-exceptional Riemann surfaceS, a closed non-empty subsetC of S, and a positive numberε. If S∗ := S \ C, then we have that1 < LS∗(γ )/LS(γ ) <

coth(ε/2), for every curveγ ⊂ S with finite length inS such thatdS(γ,C) � ε.

We need the following definitions in order to state one of our main theorems.

Definition 3.1. A normal neighborhoodof a subsetF of a Riemann surface is a compact bordeRiemann surfaceV such thatF ⊂ V, V has connection ordern (with n ∈ {1,2}) and∂V is the union ofn closed curves.

A set E = ⋃n En in an open non-exceptional Riemann surfaceS, with {En}n compact simply

connected sets, is called(r, s)-uniformly separatedin S if there exist normal neighborhoodsVn of En

such thatdS(∂Vn,En) � r , LS(∂Vn) � s for everyn, anddS(Vn,Vm) � r for everyn �= m (if ∂Vn is notconnected, byLS(∂Vn) we mean the sum of the lengths of the connected components of∂Vn).

Definition 3.2. Let S be an open non-exceptional Riemann surface,E = ⋃n En a (r, s)-uniformly

separated set inS andS∗ := S \ E. For each choice of{Vn}n we define

DS = DS

({Vn}n

) := supn

{dS |Vn

(η1

n, η2n

): η1

n, η2n are the connected components of∂Vn

andS \ ηjn is connected forj = 1,2

},

DS∗ = DS∗({Vn}n

) := supn

{dS∗ |Vn\En

(η1

n, η2n

): η1

n, η2n are the connected components of∂Vn

andS \ ηjn is connected forj = 1,2

}.

Lemma 3.1. Let S be an open non-exceptional Riemann surface andE = ⋃n En a (r, s)-uniformly

separated set inS. Let us assume that we can choose the sets{Vn}n such thatDS({Vn}n) = ∞(respectivelyDS∗({Vn}n) = ∞). ThenS (respectivelyS∗) is not hyperbolic.

Proof. For each positive integerk, we can chooseVnksuch that∂Vnk

has two connected componenη1

k, η2k , with S \ ηi

k connected anddS |Vnk(η1

k, η2k) � 4k.

SinceLS(η1k ∪ η2

k) � s, Theorem 2.2gives that there exists a(1,2s)-quasigeodesic triangle whichδ-thin with δ � k. ThenLemma Bgives thatS is not hyperbolic.

We have a similar result forS∗, sinceLS∗(η1k ∪ η2

k) � LS(η1k ∪ η2

k)coth(r/2) � s coth(r/2) (conditiondS(η

ik,E) � r allows to applyLemma C). �

SinceDS({Vn}n) � DS∗({Vn}n) by Lemma C, we deduce the following result.

Corollary 3.1. Let S be an open non-exceptional Riemann surface andE = ⋃n En a (r, s)-uniformly

separated set inS. Let us assume that we can choose the sets{Vn}n such thatDS({Vn}n) = ∞. ThenS

andS∗ are not hyperbolic.

The next result allows, in many cases, to forget punctures and funnels in order to stuhyperbolicity of a Riemann surface; this fact can be a significant simplification in the topologysurface, and therefore makes easier the study of its hyperbolicity. Recall that to delete eachEn which is(respectively, is not) an isolated point gives a puncture (respectively, a funnel) inS∗.


nd

ds

e

ereddtstzre

t

s.

nn

ave

Let us remark that we consider simply connected setsEn since we are interested in punctures afunnels. However, the condition “En is simply connected” is essentially equivalent to “En is connected”:we can assume that there is no non-trivial simple closed curveσ in En, since it is rather artificial toconsiderS∗ as a subset of a surfaceS with more topological obstacles thanS∗.

Theorem 3.2. Let S be an open non-exceptional Riemann surface andE = ⋃n En a (r, s)-uniformly

separated set inS. Then,S∗ := S \ E is δ∗-hyperbolic if and only ifS is δ-hyperbolic andDS∗({Vn}n) isfinite. Furthermore, ifDS∗({Vn}n) is finite,δ∗ (respectivelyδ) is a universal constant which only depenon r, s,DS∗({Vn}n) andδ (respectivelyδ∗).

Remark. Recall thatdS∗ �= dS |S∗ , since(S∗, dS∗) is a complete Riemannian manifold (the points ofE areat infinitedS∗-distance of the points ofS∗). This fact also implies that(S∗, dS∗) is geodesically complet(it is an open non-exceptional Riemann surface).

Proof. If DS∗({Vn}n) = ∞, Lemma 3.1gives thatS∗ is not hyperbolic. We see now that ifDS∗({Vn}n) <

∞, S∗ is hyperbolic if and only ifS is hyperbolic. This fact finishes the proof.Theorem Cis an important tool in this proof. In order to apply it, we need to construct bord

Riemann surfacesUn with better properties thanVn. If ∂Vn is connected or if∂Vn has two connectecomponentsη1

n, η2n, with dVn

(η1n, η

2n) � r/2, we defineUn := Vn. If ∂Vn has connected componen

η1n, η

2n, with dVn

(η1n, η

2n) < r/2, we defineUn in the following way: we choose two disjoint Lipschi

curvess1n, s

2n in Vn joining η1

n and η2n, with LS(s

jn) < r/2; sinceVn is a doubly connected set, the

exists a unique simply connected compact bordered Riemann surfaceUn ⊂ Vn with En ⊂ Un ands1n, s

2n ⊂ ∂Un ⊂ ∂Vn ∪ s1

n ∪ s2n .

It is clear thatUn is a normal neighborhood ofEn. SinceLS(sjn) < r/2 andUn ⊂ Vn, we have tha

dS(∂Un,En) � r/2=: r0, LS(∂Un) � s + r =: s0 for everyn, anddS(Un,Um) � dS(Vn,Vm) � r � r0 foreveryn �= m. ThenE is (r0, s0)-uniformly separated inS if we choose{Un}n as normal neighborhoodWe also haveD′

S∗ := DS∗({Un}n) � DS∗({Vn}n), and if∂Un has two connected componentsσ 1n , σ 2

n , thendUn

(σ 1n , σ 2

n ) � r/2= r0.Let us denote byK the set of indices of{Un}n. For eachn ∈ K , let us defineXn := Un and

X∗n := Un \ En.Let us consider the connected components{Xn}n∈J of S0 := S \ ⋃

n∈K intUn. If we defineX∗n := Xn

for n ∈ J , thenS = ⋃n∈Λ Xn andS∗ = ⋃

n∈Λ X∗n, with Λ := K ∪ J . We have that eachXn (with the

restricted metric ofS) andX∗n (with the restricted metric ofS∗) are bordered non-exceptional Riema

surfaces, for anyn ∈ Λ; hence they are geodesic metric spaces.We define the setG0 as the set of indicesn ∈ K such that∂Un has two connected componentsσ 1

n , σ 2n ,

anddS |S0(σ1n , σ 2

n ) < r0.In order to applyTheorem C, let us prove thatS and S∗ (and S∗ and S) have comparable

decompositions, given by{Xn}n∈Λ and{X∗n}n∈Λ:

(a) We haveXn∩Xm = X∗n ∩X∗

m =: ⋃i∈Inmηi

nm, where we defineηinm as follows:ηi

nm is a simple closedcurve ifn,m /∈ G0 (thenInm has at most two elements); ifn ∈ G0 or m ∈ G0, ηnm := Xn ∩Xm = X∗

n ∩X∗m

(thenInm has at most one element, althoughηnm can have two connected components: we do not hany hypothesis about the connection ofηi

nm in Definitions 2.7 and 2.8).Any geodesic segment inS meets at most a finite number ofηi

nm’s, sincedS(Ua,Ub) � r for anya, b ∈ K with a �= b. The same result is true inS∗.


.

e

k

to

face

face

d

(b) Lemma Cguarantees that diamXn(ηi

nm) � diamX∗n(ηi

nm) � s0 coth(r0/2), if n,m /∈ G0; we also havediamXn

(ηnm) � diamX∗n(ηnm) � LXn

(∂Un)coth(r0/2) + D′S∗ � s0 coth(r0/2) + D′

S∗ , if n ∈ G0 or m ∈ G0.In fact, if n ∈ K , we have

∑m,i diamXn

(ηinm) �

∑m,i diamX∗

n(ηi

nm) � LXn(∂Un)coth(r0/2) + D′

S∗ �s0 coth(r0/2) + D′

S∗ .(c) We can splitΛ into F1 ∪ F2 ∪ G with G := G0 ∪ (K \ L), F1 := L \ G0 andF2 := J , whereL is

the set of indicesn ∈ K such thatS \ σjn is connected for some connected componentσ

jn of ∂Un (let us

observe thatS \σ 1n is connected if and only ifS \σ 2

n is connected, sinceσ 1n andσ 2

n are freely homotopic)Then:

(c1) If n ∈ G, Xn (andX∗n) is a(s0 coth(r0/2) + D′

S∗,0, s0 coth(r0/2) + D′S∗)-tree-piece, if we choos

An = ∅: if n ∈ K \ L, each connected component of∂Un disconnectsS, and consequently,�Inm � 1; ifn ∈ G0, there is just oneηnm = σ 1

n ∪ σ 2n and henceS \ ηnm = S \ {σ 1

n ∪ σ 2n } is not connected.

(c2) Let us considern ∈ F2 = J ; if m �= k, dX∗n(ηi

nm, ηj

nk) � dXn(ηi

nm, ηj

nk) � r0, sincedS(Um,Uk) � r0;if m ∈ F1, dX∗

n(η1

nm, η2nm) � dXn

(η1nm, η2

nm) = dXn(σ 1

m,σ 2m) � r0, sincem /∈ G0; if m ∈ G, there is just one

ηnm. If n ∈ F1 = L \ G0, we havedX∗n(ηi

nm, ηj

nk) � dXn(ηi

nm, ηj

nk) = dXn(σ 1

n , σ 2n ) � r0. (See the remar

afterDefinition 2.8.)(c3) If n ∈ F1 = L \ G0, we consider geodesicsγ ij

mnk andhij

mnk in Xn andX∗n respectively, joining

ηinm = σ 1

n andηj

nk = σ 2n , with LS(γ

ij

mnk) = dXn(σ 1

n , σ 2n ) � r0 andLS∗(h

ij

mnk) = dX∗n(σ 1

n , σ 2n ) � D′

S∗ ; if we

definefij

mnk as the dilatation betweenγ ij

mnk andhij

mnk , it is a(D′S∗/r0,0)-quasiisometry. We do not need

check the last condition in (c3) since∂Un has just two connected components.(c4) If n ∈ F2 = J , we have that the identityin :Xn → X∗

n is a(coth(r0/2),0)-quasiisometry byLem-ma C.

ThenS andS∗ (andS∗ andS) have comparable decompositions, given by{Xn}n∈Λ and{X∗n}n∈Λ.

For any n ∈ K , we have thatXn = Un is a compact bordered non-exceptional Riemann surwith connection ordern � 2, such that∂Un is the union ofn closed curves. SinceLS(∂Un) � s0,Proposition 3.1guarantees thatXn is k6-hyperbolic, with a constantk6 which only depends ons0.

For anyn ∈ K , we have thatX∗n = Un \ En is a compact bordered non-exceptional Riemann sur

with connection ordern + 1 � 3, such that∂Un is the union ofn closed curves. SinceLS∗(∂Un) �LS(∂Un)coth(r0/2) � s0 coth(r0/2) by Lemma C, Propositions 3.1 and 3.2guarantee thatX∗

n is k∗6-

hyperbolic, with a constantk∗6 which only depends onr0 ands0.

Let us observe thatΛ \ F2 = K . Consequently,Theorem Cgives that ifS∗ is δ∗-hyperbolic, thenS is δ-hyperbolic, whereδ only depends onr, s,DS∗ and δ∗, and that ifS is δ-hyperbolic, thenS∗ isδ∗-hyperbolic, whereδ∗ only depends onr, s,DS∗ andδ. �

Theorem 3.2has the following direct consequence.

Corollary 3.2. Let S be an open non-exceptional Riemann surface andE = ⋃n En a (r, s)-uniformly

separated set inS. Let us assume also that we can choose the sets{Vn}n such that every connectecomponent of each∂Vn disconnectsS. Then,S is δ-hyperbolic if and only ifS∗ := S \ E is δ∗-hyperbolic. Furthermore,δ∗ (respectivelyδ) is a universal constant which only depends onr, s and δ

(respectivelyδ∗).

Next we introduce a concept which will be used in the theorems below.


ite

olic. It

e

eed

nto

Definition 3.3. If c is a positive constant, we say that a non-exceptional Riemann surfaceS (with orwithout boundary) hasc-wide genusif every homotopically non-trivial simple closed curveγ ⊂ S suchthatS \ γ is connected, verifiesLS(γ ) � c. We say thatS hasnarrow genusif there is notc > 0 such thatS hasc-wide genus.

Observe that ifS is open, it hasc-wide genus if and only if every simple closed geodesicγ ⊂ S suchthatS \ γ is connected, verifiesLS(γ ) � c.

Notice that any plane domain hasc-wide genus for everyc, and that any Riemann surface with fingenus hasc-wide genus for somec.

We will need the following general criteria which guarantees that many surfaces are not hyperbis used in the proofs ofTheorem 3.4, and[26, Theorem 3.8].

Theorem 3.3. Any non-exceptional Riemann surface(with or without boundary) with narrow genus isnot hyperbolic.

Proof. Let us consider first an open non-exceptional Riemann surfaceS with narrow genus. We choosa sequence of simple closed geodesics{γn}n in S with S \ γn connected and limn→∞ LS(γn) = 0.

The Collar Lemma[21] says that there exists a collar ofγn of width d, for every 0< d � dn, wherecoshdn = coth(LS(γn)/2).

We define the bordered Riemann surfacesSn2 as the collar ofγn of width dn/2, andSn

1 := S \ Sn2, which

is connected sinceS \ γn is connected. We have that∂Sn1 = ∂Sn

2 = Sn1 ∩ Sn

2 = ηn1 ∪ ηn

2, with

LS(ηni ) = LS(γn)cosh(dn/2) = LS(γn)

√coshdn + 1

2= LS(γn)

√coth(LS(γn)/2) + 1

2.

SinceSn2 is the collar ofγn of width dn/2, we have thatdS(η

n1, η

n2) = dSn

2(ηn

1, ηn2) = dn. By Theorem 2.1,

if S is δ-thin, thenδ � dn/2−LS(ηni )/2. Since limn→∞ dn = ∞ and limn→∞ LS(η

ni ) = 0, we have thatS

is not hyperbolic.If S has boundary, it is contained in an open non-exceptional Riemann surfaceR. Let us choose simpl

closed curves{gn}n in S with S \gn connected and limn→∞ LS(gn) = 0. Let us consider the simple closgeodesicγn in R freely homotopic togn; we have thatR \ γn is connected and limn→∞ LR(γn) = 0. Eachgeodesicγn has inR a collar of widthd, for every 0< d � dn, with coshdn = coth(LS(γn)/2).

We define the bordered Riemann surfacesRn2 as the collar ofγn in R of width dn/2, Sn

2 as aconnected component ofS ∩ Rn

2 such thatS \ Sn2 is connected, andSn

1 := S \ Sn2. We have that∂Sn

1 =∂Sn

2 = Sn1 ∩ Sn

2 = ηn1 ∪ ηn

2, with LS(ηni ) � LR(γn)

√(coth(LR(γn)/2) + 1)/2, anddSn

2(ηn

1, ηn2) � dn. Since

limn→∞ dSn2(ηn

1, ηn2) = ∞ and limn→∞ LS(η

ni ) = 0, we have thatS is not hyperbolic, byTheorem 2.2and

Lemma B. �If En is a single point for everyn, Theorems 3.1, 3.2 and 3.3allow to prove a result with a stateme

much simpler thanTheorem 3.2; in fact,S∗ is hyperbolic if and only ifS is hyperbolic (we do not need tconsiderDS∗). This theorem is also an improvement of[24, Theorem 3.3], in the direction of weakeningthe hypothesis on the setE (see[24]). We need a definition.

Definition 3.4. A setE in an open non-exceptional Riemann surfaceS is calledr-uniformly separatedifthe balls{BS(p, r)}p∈E are pairwise disjoint.


alitiess(see,

n

it

atn

s

ryt

ng

tant)

,

Ther-uniformly separated sets play a central role in the study of hyperbolic isoperimetric inequin open Riemann surfaces (see[1, Theorem 1]and[8, Theorems 3 and 4]). There are interesting relationof the hyperbolic isoperimetric inequality with other conformal invariants of a Riemann surfacee.g.,[1], [6, p. 95], [8], [29, p. 333]).

Theorem 3.4. Let S be an open non-exceptional Riemann surface andE a r-uniformly separated set iS. Then,S∗ := S \ E is δ∗-hyperbolic if and only ifS is δ-hyperbolic. Furthermore,δ∗ (respectivelyδ)only depends onc, r andδ (respectivelyδ∗), wherec is the best constant such thatS hasc-wide genus.

The conclusion ofTheorem 3.4is not true without the hypothesis aboutE, even for plane domains:is sufficient to considerS := C \ {0,1} (which is hyperbolic byTheorem D) andS∗ := C \ Z2 (which isnot hyperbolic since it has an isometry group isomorphic toZ2).

Proof. We assume first thatS hasc-wide genus, for somec. For eachp, the set ofr ’s such that∂B(p, r)

is not the union of simple closed curves (that is to say,BS(p, r) is not a bordered Riemann surface) ismost countable. SinceE is at most countable, the set ofr ’s such thatBS(p, r) is not a bordered Riemansurface for somep ∈ E, is at most countable. Let us considerr0 < 1

2 min{c, r, log 3}, such thatBS(p, r0)

is a bordered Riemann surface for everyp ∈ E.We see now thatE is a (r0,2π sinhr0)-uniformly separated set inS, with normal neighborhood

Vp := BS(p, r0). We have for anyp ∈ E thatBS(p, r0) is simply or doubly connected byTheorem 3.1.Furthermore, each connected component of∂BS(p, r0) disconnectsS: This is clear ifBS(p, r0) is simplyconnected. IfBS(p, r0) is not simply connected, then there exists a non-trivial simple closed curveg inBS(p, r0) with length less than 2r0 < c, and thereforeg disconnectsS; we have the result since evenon-trivial simple closed curve inBS(p, r0) is freely homotopic tog by Theorem 3.1. We also have thadS(p, ∂BS(p, r0)) = r0 andLS(∂BS(p, r0)) � 2π sinhr0 for everyp ∈ E, anddS(BS(p, r0),BS(q, r0)) �r > r0, for everyp �= q.

HenceE is a(r0,2π sinhr0)-uniformly separated set inS, and the result follows fromCorollary 3.2.If S has narrow genus, thenTheorem 3.3guarantees thatS is not hyperbolic. The same reasoni

as above takingr1 < 12 min{r, log 3} shows thatE is a (r1,2π sinhr1)-uniformly separated set inS (the

dependence ofr0 on c is just used to prove that each connected component of∂BS(p, r0) disconnectsS).Consequently,Theorem 3.2allows to deduce thatS∗ is not hyperbolic. �

If we considerS∗ := S \ {p1,p2}, whereS is an open Riemann surface andp1,p2 ∈ S, there areseveral conformal invariants ofS∗ (e.g., the exponent of convergence and the isoperimetric conswhich degenerate whenp2 tends top1. We have the following surprising consequence ofTheorem 3.4about stability of hyperbolicity.

Corollary 3.3. LetS be aδ-hyperbolic open non-exceptional Riemann surface withc-wide genus. Thenfor each natural numbern there exists a constantδn, which only depends onδ, c and n, such thatS \ {p1, . . . , pn} is δn-hyperbolic, for anyp1, . . . , pn ∈ S.

Proof. We prove the theorem by induction onn. Theorem 3.4gives the result forn = 1 (E = {p1} is r-uniformly separated for anyr). Let us assume that the result is true forn−1; thenS∗ := S \{p1, . . . , pn−1}is δn−1-hyperbolic, for anyp1, . . . , pn−1 ∈ S.


ningrvent

n

elpful

at.

J. 49

Observe thatS∗ hasc-wide genus, sincedS\F � dS for any closed setF ⊂ S. ThenTheorem 3.4givesthat S∗ \ {pn} is δn-hyperbolic, whereδn is a constant which only depends onδn−1 andc (E = {pn} isr-uniformly separated for anyr). �

Now we give a simple condition which impliesDS∗ = ∞, just in terms of distances inS.

Definition 3.5. Let S be an open non-exceptional Riemann surface andE = ⋃n En a (r, s)-uniformly

separated set inS. For each fixed choice of{Vn}n we denote byL the set of indicesn such that∂Vn hassome connected componentηn with S \ ηn connected. Ifn ∈ L, let us denote byC(En) the set of curvesγ joining En with itself, such that inEn ∪ γ there exists a curveσ with S \ σ connected. We define

CS

({Vn}n

) := inf{LS(γ ): γ ∈ C(En) for somen

}.

Proposition 3.3. Let S be an open non-exceptional Riemann surface andE = ⋃n En a (r, s)-uniformly

separated set inS. If for some choice of the sets{Vn}n, we haveCS({Vn}n) = 0, thenDS∗({Vn}n) = ∞.

Proof. We can choosenk and a geodesicγk which has minimal length inC(Enk), with LS(γk) =

4εk < r and limk→∞ εk = 0. If we consider the universal covering, we see that any curve joithe two connected components of∂Vnk

in S∗ is longer or equal lengthed than the shortest cugk in D \ {i tanhεk,−i tanhεk} joining {|z| = tanh(r/4)} with itself and intersecting the segmejoining i tanhεk and −i tanhεk (let us observe thatdD(−i tanhεk, i tanhεk) = 2dD(0, i tanhεk) = 4εk ,by (3.1)). It is not difficult to see thatgk is the interval[− tanh(r/4), tanh(r/4)]. ThenDS∗({Vn}n) �supk LD\{i tanhεk,−i tanhεk}([− tanh(r/4), tanh(r/4)]). We denote byDt the disk with center 0 and Euclidearadiust . Since

limk→∞LD\{i tanhεk,−i tanhεk}

([− tanh(r/4), tanh(r/4)])

= limk→∞ LDcothεk

\{i,−i}([− tanh(r/4)cothεk, tanh(r/4)cothεk

])= LC\{i,−i}

((−∞,∞)

) = ∞,

we have thatDS∗({Vn}n) = ∞. �

Acknowledgement

We would like to thank the referee for his/her careful reading of the manuscript and for some hsuggestions.

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