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Stability of Gromov hyperbolicity.

1. INTRODUCTION

The theory of Gromov hyperbolic spaces is a useful tool in order to understand the connections between graphs and Potential Theory (see e.g. [4], [11], [14], [27], [28], [29], [30], [37], [38], [43]). Besides, the concept of Gromov hyperbolicity grasps the essence of negatively curved spaces, and has been successfully used in the theory of groups (see e.g. [16], [18], [19] and the references therein).

A geodesic metric space is called hyperbolic (in the Gromov sense) if there exists an upper bound of the distance of every point in a side of any geodesic triangle to the union of the two other sides (see Definition 2.1). The latter condition is known as Rips condition.

But, it is not easy to determine whether a given space is Gromov hyperbolic or not. Recently, there has been some research aimed to show that metrics used in geometric function theory are Gromov hyperbolic. Some specific examples are showing that the Klein-Hilbert metric ([8], [31]) is Gromov hyperbolic (under particular conditions on the domain of Definition), that the Gehring-Osgood metric ([21]) is Gromov hyperbolic, and that the Vuorinen metric ([21]) is not Gromov hyperbolic (except for a particular case). Recently, some interesting results by Balogh and Buckley [5] about the hyperbolicity of Euclidean bounded domains with their quasihyperbolic metric have made significant progress in this direction (see also [10], [44] and the references therein). Another interesting instance is that of a Riemann surface endowed with the Poincare metric. With such metric structure a Riemann surface is always negatively curved, but not every Riemann surface is Gromov hyperbolic, since topological obstacles may impede it: for instance, the two-dimensional jungle-gym (a [Z.sup.2]-covering of a torus with genus two) is not hyperbolic. We are interested in studying when Riemann surfaces equipped with their Poincare metric are Gromov hyperbolic (see e.g. [22], [23], [24], [25], [40], [41], [42], [32], [33], [34], [3], [36]).

One of the important problems when studying any property is to obtain its stability under appropriate deformations, i.e., under what type of perturbations it is preserved. With this aim, here we study the stability of Gromov hyperbolicity.

First of all we analyze the stability under appropriate limits, in the context of general metric spaces (see Theorem 3.1), and we apply this result to plane domains with their Poincare metrics and Euclidean domains with their quasihyperbolic metrics (see respectively Theorems 3.3 and 3.4). We also have complementary results on stability of non-hyperbolicity for the Poincare and quasihyperbolic metrics (see respectively Theorems 3.6 and 3.5).

We also prove the stability of Gromov hyperbolicity under some transformations in plane domains (endowed with their Poincare metrics), even though the original domain and the modified one are not quasi-isometric. In particular, Theorem 5.1 gives some answers to the following question: how do some geometric perturbations affect on the hyperbolicity of a flute surface?

Notations. We denote by X a geodesic metric space. By [d.sub.X] and [L.xub.X] we shall denote, respectively, the distance and the length in the metric of X. From now on, when there is no possible confusion, we will not write the subindex X.

Finally, we denote by C, c and [c.sub.i], positive constants which can assume different values in different theorems.

2. BACKGROUND IN GROMOV SPACES AND RIEMANN SURFACES

In our study of hyperbolic Gromov spaces we use the notations of [16]. We give now the basic facts about these spaces. We refer to [16] for more background and further results.

Definition 2.1. If X is a geodesic metric space and J = {[J.sub.1], [J.sub.2],...,[J.sub.n]}, with [J.sub.j] [subset or equal to] X, we say that J is [delta]-thin if for every x [member of] [J.sub.1] we have that d [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [x.sub.1],[x.sub.2],[x.sub.3] [member of] X, a geodesic triangle T = {[x.sub.1],[x.sub.2],[x.sub.3]} is the union of three geodesics [[x.sub.1],[x.sub.2]], [[x.sub.2],[x.sub.3]] and [[x.sub.3],[x.sub.1]]. The space X is [delta]-hyperbolic (or satisfies the Rips condition with constant [delta]) if every geodesic triangle in X is [delta]-thin.

We would like to point out that deciding whether or not a space is hyperbolic is usually extraordinarily difficult: Notice that, first of all, we have to consider an arbitrary geodesic triangle T, and calculate the minimum distance from an arbitrary point P of T to the union of the other two sides of the triangle to which P does not belong to. And then we have to take supremum over all the possible choices for P and then over all the possible choices for T. It means that if our space is, for instance, an n-dimensional manifold and we select two points P and [OMEGA] on different sides of a triangle T, the function F that measures the distance between P and Q is a (3n + 2)-variable function. In order to prove that our space is hyperbolic we would have to take the minimum of F over the variable that describes Q, and then the supremum over the remaining 3n + 1 variables, or at least prove that it is finite. Without disregarding the difficulty of solving a (3n + 2)-variable minimax problem, notice that the main obstacle is that we do not even know in an approximate way the location of geodesics in the space.

Examples:

(1) Every bounded metric space X is (diam X)-hyperbolic (see e.g. [16, p. 29]).

(2) Every complete simply connected Riemannian manifold with sectional curvature which is bounded from above by -k, with k > 0, is hyperbolic (see e.g. [16, p. 52]).

(3) Every tree with edges of arbitrary length is 0-hyperbolic (see e.g. [16, p. 29]).

Definition 2.2. If [gamma]: [a,b] [right arrow] X is a continuous curve in a metric space (X, d), the length of [gamma] is

L([gamma]):=sup{[n.summation over (i=1)]d([gamma]([t.sub.i-1]), [gamma]([t.sub.i])): a = [t.sub.0] [less than or equal to] [t.sub.1] [less than or equal to] ... [less than or equal to] [t.sub.n] = b}.

We say that [gamma] is a geodesic if it is an isometry, i.e. L([[gamma]|.sub.[t,s]]) = d([gamma](t), [gamma](s)) = [absolute value of t-s] for every s, t [member of] [a, b]. We say that X is a geodesic metric space if for every x, y [member of] X there exists a geodesic joining x and y; we denote by [x, y] any of such geodesics (since we do not require uniqueness of geodesics, this notation is ambiguous, but convenient as well).

If E is a relatively closed subset of a geodesic metric space (X, d), we always consider in E the inner metric obtained by the metric in X, that is

[d.sub.X|E](z,w):=inf {[L.xub.X,d]([gamma]) : [gamma] [subset] E is a rectifiable curve joining z and w} [greater than or equal to] d(z,w).

Definition 2.3. A function between two metric spaces f: X [right arrow] Y is an (a, b)-quasi-isometry, a [greater than or equal to] 1, b [greater than or equal to] 0, if

1/a [d.sub.X]([x.sub.1],[x.sub.2]) - b [less than or equal to] dY(f([x.sub.1]), f([x.sub.2])) [less than or equal to] a [d.sub.X]([x.sub.1],[x.sub.2]) + b, for every [x.sub.1],[x.sub.2] [member of] X.

We say that f is [epsilon]-full if for every y [member of] Y there exists x [member of] X with dY(y, f (x)) [less than or equal to] e. If f is [epsilon]-full for some [epsilon] [greater than or equal to] 0, we say that X and Y are quasi-isometric.

An (a, b)-quasigeodesic in X is an (a, b)-quasi-isometry between an interval of R and X.

Quasi-isometries are important since they are the maps which preserve hyperbolicity:

Theorem 2.1. ([16, p. 88]) Let us consider an (a,b)-quasi-isometry between two geodesic metric spaces f: X [right arrow] Y. If Y is [delta]-hyperbolic, then X is [delta]'-hyperbolic, where [delta]' is a constant which only depends on [delta], a and b. Besides, if the image of f is [epsilon]-full for some [epsilon] [greater than or equal to] 0, then X is hyperbolic if and only if Y is hyperbolic.

It is well-known that if f is not [epsilon]-full, the hyperbolicity of X does not imply the hyperbolicity of Y: it is enough to consider the inclusion of R in [R.sup.2]] (which is indeed an isometry).

Definition 2.4. Let us consider H > 0, a metric space (X, d), and subsets Y, Z [subset of equal to] X. The set [V.sub.H](Y) := {x [epsilon] X : d(x,Y) [less than or equal to] H} is called the H-neighborhood of Y in X. The Hausdorff distance of Y to Z is defined by

[H.sub.d](Y, Z) := inf{H > 0 : Y [subset of equal to] [V.sub.H](Z), Z [subset or equal to] [V.sub.H](Y)}.

The following is a beautiful and useful result:

Theorem 2.2. ([16, p. 87]) For each [delta] [greater than or equal to] 0, a [greater than or equal to] 1 and b [greater than or equal to] 0, there exists a constant H = H([delta], a, b) with the following property:

Let (X, d) be a [delta]-hyperbolic geodesic metric space and let g be a (a, b)-quasigeodesic joining x and y. If [gamma] is a geodesic joining x and y, then [H.sub.d](g, [gamma]) [less than or equal to] H.

This property is known as geodesic stability. M. Bonk has proved that, in fact, geodesic stability is equivalent to Gromov hyperbolicity [9].

Definition 2.5. Let (X, d) be a metric space, and let {[X.sub.n]}n [subset or equal to] X be a family of geodesic metric spaces such that [[eta].sub.nm] := [X.sub.n] [intersection] [X.sub.m] are compact sets. Further, assume that for any n and m the set X \ [[eta].sub.nm] is not connected, and that a and b are in different components of X [[eta].sub.nm] for any a [member of] [X.sub.n] \ [[eta].sub.nm], b [member of] [X.sub.m] \ [[eta].sub.nm], with m [not equal to] n. If there exists positive constants [c.sub.1] and [c.sub.2] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([[eta].sub.nm]) [less than or equal to] [c.sub.1] for every n, m, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]([[eta].sub.nm], [[eta].sub.nk]) > [c.sub.2] for every n and m [not equal to] k, we say that [{[X.sub.n]}.sub.n] is a ([c.sub.1],[c.sub.2])-tree decomposition of X.

Theorem 2.3. ([40, Theorem 2.4] and [32, Theorem 2.9]) Let us consider a metric space X and a family of geodesic metric spaces [{[X.sub.n]}.sub.n] [subset or equal to] X which is a ([c.sub.1], [c.sub.2])-tree decomposition of X. Then X is hyperbolic if and only if there exists a constant [[delta].sub.0] such that [X.sub.n] is [[delta].sub.0]-hyperbolic for every n.

A non-exceptional Riemann surface S is a Riemann surface whose universal covering space is the unit disk D = {z [member or] C : [absolute value of z] < 1}, endowed with its Poincare metric, i.e. the metric obtained by projecting the Poincare metric of the unit disk ds = 2[absolute value of dz]/(1 -[[absolute value of z].sup.2]). Therefore, any simply connected subset of S is isometric to a subset of D. With this metric, S is a geodesically complete Riemannian manifold with constant curvature -1, and therefore S is a geodesic metric space. The only Riemann surfaces which are left out are the exceptional Riemann surfaces, that is to say, the sphere, the plane, the punctured plane and the tori. It is easy to study the hyperbolicity of these particular cases. The Poincare metric is natural and useful in Complex Analysis: for instance, any holomorphic function between two domains is Lipschitz with constant 1, when we consider the respective Poincare metrics.

For x [member of] D [??] [??] we denote by [[delta].sub.D] (x) the distance of x to the boundary of D, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The quasihyperbolic metric in a domain D [??][R.sup.k] is the distance induced by the density 1/[[delta].sub.D](x). We will denote by [partial derivative]D the quasihyperbolic or the Poincare distance in D, indistinctly; the context will determine without a doubt which metric we are using each time. The subscript Eucl will be used to denote the distance or length with respect to the Euclidean metric.

It is well known that for every non-exceptional domain [OMEGA] [subset] C

[[lambda].sub. [OMEGA] (z) [less than or equal to] 2/[[delta].sub. [OMEGA] , [for all] z [member of] [OMEGA] ,

and that for all domains [[OMEGA].sub.1] [subset] [[OMEGA].sub.2] we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every z [member of] [[OMEGA].sub.1].

We will need the following result.

Theorem 2.4. ([22, Theorem 1.1]) Let [OMEGA] be a plane domain with [delta][OMEGA] [subset] R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(1) The metric spaces [OMEGA], with either the Poincare or the quasihyperbolic metric, are Gromov hyperbolic if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) The metric spaces Q, with either the Poincare or the quasihyperbolic metric, are not Gromov hyperbolic if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. LIMITS OF GROMOV HYPERBOLIC SPACES

First of all we analyze the stability of Gromov hyperbolicity under appropriate limits, in the context of general metric spaces.

Theorem 3.1. Let us consider geodesic metric spaces X and {[X.sub.n]}n, with X, [X.sub.n] [subset or equal to] Y, where Y is a space with a measure [mu] and functions [lambda], [[lambda].sub.n], such that [L.xub.X]([gamma]) = [[integral].sub.[gamma]] [[lambda].sub.n] d[mu], for every curve [gamma] in X and [X.sub.n] respectively. We assume also that for each ball B of X there is a positive constant [c.sub.B] with [lambda] [greater than or equal to] [c.sub.B] in B. Let us assume that for every closed ball B [subset] X there exists N with B [subset or equal to] [X.sub.n] for every n [greater than or equal to] N and [[lambda].sub.n] converges to [lambda] uniformly in B. If there exists a constant [[delta].sub.0] such that [X.sub.n] is [[delta].sub.0]-hyperbolic for every n, then X is [delta]-hyperbolic, with [delta] is a constant which just depends on [[delta].sub.0].

Proof. Given [epsilon] > 0 and a closed ball B [subset or equal to] X, there exists N with B [subset or equal to] [X.sub.n] and [absolute value of [[lambda]sub.n]-[lambda]] [less than or equal to] [ec.sub.B] in B for n [greater than or equal to] N. Observe that in B we have that [absolute value of [[lambda].sub.n]/[lambda] - 1]][less than or equal to] [ec.sub.B]/[lambda] [less than or equal to][epsilon].

Therefore we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.1)

for n [greater than or equal to] N and for every curve [gamma] [subset or equal to] B.

First of all we prove that for each p [member of] X and 0 < [r.sub.1] < [r.sub.2] < [r.sub.3], there exists N with B [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We show now the first content. Let us consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For any x [member of] [partial derivative] [B.sub.X](p,[r.sub.2]) we take any curve [eta] joining p and x. By (3.1) there exists [N.sub.1] such that for n [greater than or equal to] [N.sub.1] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [eta] is a curve joining x and y in [X.sub.n] and n is not contained [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, every geodesic in [X.sub.n] joining x and y is contained in B = [B.sub.X] (p, 5r). Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for n > [N.sub.0] and for every x,y [member of] [B.sub.X](p,r). This implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges uniformly on closed balls of X to [d.sub.X].

We consider now a geodesic triangle T in X, a point p [member of] T and 0 < [member of] < 1/2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us consider g: [a, b] [right arrow] X a side of T. We check now that for any n [greater than or equal to] [N.sub.4], g is an ([[alpha].sub.[epsilon]], 0)-quasigeodesic in [X.sub.n]: if a [less than or equal to] c [less than or equal to] d [less than or equal to] b, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Theorem 2.2, if n [greater than or equal to] [N.sub.4], we have that the Hausdorff distance of g and any geodesic segment [g(a),g(b)] in [X.sub.n] is less or equal than M. Then, T is ([[delta].sub.0] + 2M)-thin in [X.sub.n] for every n [greater than or equal to] [N.sub.4], and consequently, T is [delta]-thin in X, with [delta] = ([[delta].sub.0] + 2M)/(1 - e). Since 0 < [epsilon] < 1/2, we have that T is ([[delta].sub.0] + 2M)-thin in X. []

We want to apply this result to the context of Euclidean domains. In order to do it, we will need some definitions.

Definition 3.1. Let [{[D.sub.n]}.sup.[infinity].sub.n=1] be a sequence of domains in the Riemann sphere such that [z.sub.0] [member of] [D.sub.n], for some fixed point [z.sub.0] in the Riemann sphere. The kernel of the sequence [{[D.sub.n]}.sup.[infinity].sub.n=1] with respect to [z.sub.0], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is defined as the largest domain D such that: (a) [z.sub.0] [member of] D; (b) for each compact subset K of D, K C [D.sub.n] for all n sufficiently large. It is simple to check that the definition makes sense and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is nonvoid if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] contains some neighborhood of [z.sub.0].

We say that [D.sub.n] [right arrow] D if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every subsequence {[n.sub.j]} of {n}.

Definition 3.2. Let [OMEGA] be any domain in the Riemann sphere such that [infinity] [member of] [OMEGA] and card([partial derivative][OMEGA]) [greater than or equal to] 3. The normalized universal covering map [pi]: D [right arrow] [OMEGA] is the unique universal covering map with [pi](z) [approximately equal to] c [z.sup.-1], c > 0, as z [right arrow] 0.

The following is a result of Hejhal (see [26]); in fact, the result in [26] is better, but this version is good enough for the application that we need.

Theorem 3.2. ([26, Theorem 1]) Let [{[D.sub.n]}.sup.[infinity].sub.n=1] be a sequence of domains such that [infinity] [member of] [D.sub.n] and card([partial derivative][D.sub.n]) [greater than or equal to] 3. In addition, let [[pi].sub.n] be the normalized universal covering map for [D.sub.n]. If D = [ker.sub.[infinity]]{[D.sub.n]} is nonvoid, card([partial derivative]D) [greater than or equal to] 3 and [D.sub.n] [right arrow] D, then [[pi].sub.n] converge uniformly on compact sets to the normalized covering map for D.

Theorems 3.1 and 3.2 give the following result for plane domains endowed with their Poincare metrics.

Theorem 3.3. Let [{[D.sub.n]}.sup.[infinity].sub.n=1] be a sequence of domains such that [z.sub.0] [member of] [D.sub.n], card([partial derivative][D.sub.n]) [greater than or equal to] 3 and [D.sub.n] endowed with its Poincare metric is [[delta].sub.0]-hyperbolic for every n. If D = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is nonvoid, card([partial derivative]D) [greater than or equal to] 3 and [D.sub.n] [right arrow] D, then D endowed with its Poincare metric is [delta]-hyperbolic, where [delta] is a constant which just depends on [[delta].sub.0].

Proof. Applying a Mobius map if it is necessary (which is an isometry for the Poincare metric), we can assume that [z.sub.0] = [infinity]. The Poincare density [[lambda].sub.n] of [D.sub.n] only depends on the universal covering map and its derivative. Then, Theorem 3.2 gives that [[lambda].sub.n] converges to [lambda] uniformly on closed balls of D. Since [lambda] is a positive continuous function, for each ball of D there is a positive constant c with [lambda] [greater than or equal to] c. Consequently, the result holds by Theorem 3.1. []

We need the following standard result.

Proposition 3.1. Let X,Y be metric spaces with X compact, and let {[f.sub.n]}n be a sequence of functions [f.sub.n]: X [right arrow] Y verifying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = f (x) for every x [member of] X, and dY([f.sub.n](x),[f.sub.n](y)) [less than or equal to] M[d.sub.X](x,y) for every x,y [member of] X and every n. Then {[f.sub.n]}n converges uniformly to f on X.

Proof. Note that dY (f(x), f(y)) [less than or equal to] M[d.sub.X](x,y) for every x,y [member of] X. Assume that {[f.sub.n]} does not converge uniformly to f. Then there exist [epsilon] > 0, {[x.sub.j]} [member of] X and {[n.sub.j]} with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every j. Since X is compact, without loss of generality we can assume that {[x.sub.j]} converges to x [member of] X. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. []

Now we can obtain a consequence of Theorem 3.1 for the quasihyperbolic metric.

Let [{[D.sub.n]}.sup.[infinity].sub.n=1] be a sequence of domains in [??] such that [x.sub.0] [member of] [D.sub.n] for some fixed point [x.sub.0] [member of] [??], and [D.sub.n] [not equal to] [??]. We can define the kernel of the sequence [{[D.sub.n]}.sup.[infinity].sub.n=1] with respect to [x.sub.0], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in a similar way than in the case of the Riemann sphere.

Theorem 3.4. Let [{[D.sub.n]}.sup.[infinity].sub.n=1] be a sequence of domains such that [x.sub.0] [member of] [D.sub.n], [D.sub.n] [not equal to] [??] and [D.sub.n] endowed with its quasihyperbolic metric is [[delta].sub.0]-hyperbolic for every n. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [D.sub.n] [right arrow] D, then D endowed with its quasihyperbolic metric is [delta]-hyperbolic, where [delta] is a constant which just depends on [[delta].sub.0].

Proof. Note that we just need to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every x [member of] D, since if this holds, then Proposition 3.1 gives the uniform convergence on compact subsets of D (recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every x,y [member of] [D.sub.n] and every n), and therefore the result is a direct consequence of Theorem 3.1.

Let us show the pointwise convergence of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For any x [member of] D and [epsilon] > 0 there exists N with [B.sub.Eucl](x, (1 - [epsilon])[delta]D(x)) [subset] [D.sub.n] for every n [greater than or equal to] N. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Seeking for a contradiction, let us assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some x [member of] D. Therefore, there exists a subsequence {[n.sub.j]} of {n} with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a contradiction. Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every x [member of] D, and this finishes the proof.

After these results on stability of Gromov hyperbolicity under limits, we are interested in similar results on stability of non-hyperbolicity under limits.

First of all, we show with the following example that the limit of non-hyperbolic spaces can be hyperbolic.

Example 3.1. Let us consider an increasing sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [D.sub.n] endowed either with the Poincare or the quasihyperbolic metric, are not Gromov hyperbolic by Theorem 2.4. However [D.sub.n] [right arrow] D := C\({0}[[union]{1}), and D is hyperbolic with both metrics (see e.g. [22, Proposition 3.5]).

The limit in this case does not preserve the non-hyperbolicity since "the obstacles for the hyperbolicity escape to infinity". One can think that if the topological obstacles of [D.sub.n] grow (i.e. [[product].sub.1]([D.sub.n]) is a subgroup of [[product].sub.1] ([D.sub.n+1]) for every n) and [D.sub.n] are not hyperbolic for every n, then D will not be hyperbolic.

In fact, we have a stronger result which can be stated in an informal way as follows: in order to get the non-hyperbolicity of D, it is sufficient to require the non-hyperbolicity for just one [D.sub.n], for both the quasihyperbolic and the Poincare metric.

Theorem 3.5. Let D [not equal to] [??], [??] and let W be a bounded domain in [??] with [partial derivative]W a hypersurface contained in D. If [D.sub.0] := D [union] ([??] \ W) endowed with its quasihyperbolic metric is not hyperbolic, then D endowed with its quasihyperbolic metric is not hyperbolic either.

Proof. Since [partial derivative]W is a compact set that does not intersect [partial derivative]D, we have [d.sub.Eucl]([partial derivative]W, [partial derivative]D\ [partial derivative][D.sub.0]) =: [c.sub.0] > 0.

We always have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every x [member of] D.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for every x [member of] D n [bar.W].

Let us define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that [partial derivative]W is a compact hypersurface contained in D and that every hypersurface is connected; therefore, [X.sub.2] and [Y.sub.2] are path-connected sets and geodesic metric spaces. It is clear that {[X.sub.1],[X.sub.2]} is a ([c.sub.1],[c.sub.2])-tree decomposition of D for any [c.sub.2], and that {[Y.sub.1], [Y.sub.2]} is a ([k.sub.1], [k.sub.2])-tree decomposition of [D.sub.0] for any [k.sub.2].

Since [D.sub.0] is not hyperbolic, Theorem 2.3 gives that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not hyperbolic.

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not hyperbolic by Theorem 2.1. Hence, D is not hyperbolic by Theorem 2.3.

We have a similar result for the Poincare metric. We need a previous result from [2].

Lemma 3.1. ([2, Lemma 3.1]) Let [D.sub.0] be a plane domain, let E be a closed nonempty subset of [D.sub.0], D := [D.sub.0] \ E and [epsilon] a positive constant. Then we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.6. Let D [subset] C and let W be a bounded Jordan domain in C with [partial derivative]W [subset] D. If [D.sub.0] := D U (C \ W) endowed with its Poincare metric is not hyperbolic, then D endowed with its Poincare metric is not hyperbolic either.

Proof. Note that D = [D.sub.0] \ E, with E := C \ (D [union] W). Let us define [epsilon] := min[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then Lemma 3.1 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for every z [member of] [D.sub.0] [intersection] [bar.W]. Now, a similar argument to the one in the proof of Theorem 3.5, using Theorem 2.3, finishes the proof.

4. BACKGROUND IN DENJOY DOMAINS

From now on, we will make extensive use of a specially interesting kind of Riemann surfaces, endowed with their Poincare metrics: the Denjoy domains, that is to say, plane domains [OMEGA] with [delta][OMEGA] [subset] R. This kind of surfaces are becoming more and more important in Geometric Theory of Functions, since, on the one hand, they are a very general type of Riemann surfaces, and, on the other hand, they are more manageable due to its symmetry. For instance, Garnett and Jones have proved in [15] the Corona Theorem for Denjoy domains, and in [2] and [39] the authors have got characterizations of Denjoy domains which satisfy a linear isoperimetric inequality. See also [1], [3] and [17].

We will consider a particular type of Denjoy domain, which we will call train. A train can be defined as the complement of a sequence of ordered closed intervals (see Definition 4.1). Trains do include a especially important case of surfaces which are the flute surfaces (see, e.g. [6], [7]). These ones are the simplest examples of infinite ends, and besides, in a flute surface it is possible to give a fairly precise description of the ending geometry (see, e.g. [20]). In [3] there are some partial results on hyperbolicity of trains.

We need some definitions and background. So far we have used the word geodesic in the sense of Definition 2.2, that is to say, as a global geodesic or a minimizing geodesic; however, we need now to deal with a special type of local geodesics: simple closed geodesics, which obviously can not be minimizing geodesics. We will continue using the word geodesic with the meaning of Definition 2.2, unless we are dealing with closed geodesics.

Definition 4.1. A train is a Denjoy domain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], such that -[infinity] [less than or equal to] [a.sub.0] and [b.sub.n] [less than or equal to] [a.sub.n+1] for every n. A flute surface is a train with [b.sub.n] = [a.sub.n+1] for every n.

For each n > 0, we denote by [[gamma].sub.n] the simple closed geodesic which just intersects R in ([a.sub.0], [b.sub.0]) and ([a.sub.n] [b.sub.n]), [2l.sub.n] := [L.sub.[OMEGA]([[gamma].sub.n]).

For each n > 0, we denote by [[sigma].sub.n] the simple closed geodesic which just intersects R in ([a.sub.n],[b.sub.n]) and ([a.sub.n+1], [b.sub.n+1]), and [2r.sub.n] := [L.sub.[OMEGA]] ([[sigma].sub.n]) (see figure below). If [b.sub.n] = [a.sub.n+1], we define [[sigma].sub.n] as the puncture at this point and [r.sub.n] = 0.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 4.1. Recall that in every free homotopy class there exists a single simple closed geodesic, assuming that punctures are simple closed geodesics with length equal to zero. That is why both [[gamma].sub.n] and [[sigma].sub.n] are unique for every n.

A train is a flute surface if and only if every [[sigma].sub.n] is a puncture, i.e., if [a.sub.n+1] = [b.sub.n] for every n [greater than or equal to] 0.

It is not difficult to see that the values of {[l.sub.n]} and {[r.sub.n]} determine a train. Then, there must exist a characterization of hyperbolicity in terms of the lengths of these sequences (see [35, Theorem 3.2]). This theorem has several interesting consequences.

Proposition 4.1. ([35, Proposition 3.6]) Let us consider a train [OMEGA] with [l.sub.n] [less than or equal to] c for every n. Then [OMEGA] is [delta]-hyperbolic, where [delta] is a constant which only depends on c.

The following result shows that hyperbolicity is stable under bounded perturbations of the lengths of [{[[gamma].sub.n]}.sub.n] and [{[[sigma].sub.n]}.sub.n].

Theorem 4.1. ([35, Theorem 3.8]) Let us consider two trains [OMEGA], [OMEGA]' and a constant c such that [absolute value of [r'.sub.n]- [r.sub.n]] [less than or equal to] c, and [absolute value of [l'.sub.n]- [l.sub.n]] [less than or equal to] c for every n [greater than or equal to] 1. Then [OMEGA] is hyperbolic if and only if [OMEGA]' is hyperbolic.

Furthermore, if [OMEGA] is [delta]-hyperbolic, then [OMEGA]' is [OMEGA]'-hyperbolic, with [delta]' a constant which only depends on [delta] and c.

Theorem 4.2 below is a simpler version of [35, Theorem 3.2]. Next, we are going to define some functions that will appear in the statement of Theorem 4.2.

Definition 4.2. Let us consider a sequence of positive numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a sequence of non-negative numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Consider n [greater than or equal to] 1 and 0 [less than or equal to] h [less than or equal to] [l.sub.n]. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The functions [[GAMMA].sup.0.sub.nm](h) are naturally associated to trains by taking [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 4.2. ([35, Theorem 3.12]) Let us consider a train [OMEGA] such that there exists a constant c > 0 with [r.sub.n] [less than or equal to] c for every n [greater than or equal to] 1. Then [OMEGA] is hyperbolic if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, if [OMEGA] is [delta]-hyperbolic, then [K.sup.0] is bounded by a constant which only depends on [delta] and c; if [K.sup.0] < [infinity], then [OMEGA] is [delta]-hyperbolic, with 5 a constant which only depends on [K.sup.0] and c.

We also have the following facts.

Proposition 4.2. ([35, Proposition 3.13]) In any train [OMEGA] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for every n [greater than or equal to] 1 and 0 [less than or equal to] h [less than or equal to] [l.sub.n].

Theorem 4.3. ([35, Corollary 3.14]) Let us consider a train [OMEGA] with [l.sub.1] [less than or equal to] [l.sup.0], [r.sub.n] [less than or equal to] [c.sub.1] for every n and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.1)

Then [OMEGA] is [delta]-hyperbolic, where [delta] is a constant which only depends on [c.sub.1], [c.sub.2] and [l.sup.0].

5. TRANSFORMATIONS WHICH PRESERVE HYPERBOLICITY

In the current paper our main aim is to study the stability of Gromov hyperbolicity. Keeping that in mind, from now on we will adopt a different viewpoint: we will study under what kind of geometric perturbations Gromov hyperbolicity is preserved.

To be more precise, we want to study now the following problem: if we have an hyperbolic train with {[r.sub.n]} [member of] [l.sup.[infinity]] what kind of perturbations are allowed on {[l.sub.n]} so that the train is still hyperbolic? Theorem 5.1 below answers this question, and furthermore provides methods to construct a great deal of hyperbolic flute surfaces.

We need the following definitions.

Definition 5.1. We denote by H the following set of sequences:

H : = {{[x.sub.n]} : the train with [l.sub.n] = [x.sub.n] and [r.sub.n] = 0 for every n is hyperbolic}

= {{[x.sub.n]} : every train with [l.sub.n] = [x.sub.n] for every n and {[r.sub.n]} [member of] [l.sup.[infinity]] is hyperbolic } .

The second equality is a direct consequence of Theorem 4.1.

Definition 5.2. We say that the sequence {[y.sub.n]} is a union of the sequences {[x.sup.1.sub.n]},..., {[x.sup.N.sub.n]}, if {[x.sup.1.sub.n]},..., {[x.sup.N.sub.n]} are subsequences of {[y.sub.n]}, and {[x.sup.1.sub.n]},..., {[x.sup.N.sub.n]} is a partition of {[y.sub.n]}.

Theorem 5.1. Let us consider a sequence { [l.sub.n]} [member of] H.

(1) If [l'.sub.n] = [l.sub.n] + [x.sub.n] with {[x.sub.n]} [member of] [l.sup.[infinity]], then {[l.sub.n]} [member of] H.

(2) Fix a positive integer N. Let us assume that {[l.sub.n]} is a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} of {[l'.sub.n]} such that [n.sub.k+1] - [n.sub.k] [less than or equal to] N for every k, and max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every m [member of] ([n.sub.k], [n.sub.k+1]) and every k. Then {[l'.sub.n]} [member of] H.

(3) If {[l'.sub.n]} is any union of the sequences {[l.sup.1.sub.n]},..., {[l.sup.N.sub.n]} [member of] H, then {[l'.sub.n]} [member of] H.

(4) If {[l'.sub.n]} is a union of {[l.sub.n]} and a sequence {[x.sub.n]} [member of] [l.sup.[infinity]], then {[l'.sub.n]} [member of] H.

(5) Let us assume that {[l'.sub.n]} is any union of the sequences {[l.sup.1.sub.n]},...,{l.sup.N.sub.n]} which verify

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then {[l'.sub.n]} [member of] H.

(6) Fix a positive integer N. Let us assume that {[x.sub.n]} is a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of {[l'.sub.n]} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every m [member of] ([n.sub.k], [n.sub.k+1]) and every k. If {[x.sub.n]} [??] H, then {[l'.sub.n]} [??] H.

(7) Fix a positive integer N. Let [sigma] be a permutation of the positive integer numbers such that [absolute value of [sigma](n) -n] [less than or equal to] N for every n, and consider [l'.sub.n] := [l.sub.[sigma](n)]. Then {[l'.sub.n]} [member of] H.

Remark 5.1. In fact, (7) gives the following stronger statement: If [sigma] is a permutation of the positive integer numbers such that [absolute value of [sigma](n) -n] [less than or equal to] N for every n, then {[l.sub.[sigma](n)]} [member of] H if and only if {[l.sub.n]} [member of] H (since [[sigma].sup.-1] also satisfies [absolute value of [[sigma].sup.-1](n)-n] [less than or equal to] N for every n).

Remark 5.2. We have examples showing that the conclusions of Theorem 5.1 do not hold if we remove any of the hypothesis.

Proof. (1) is a direct consequence of Theorem 4.1.

(2) Fix n [greater than or equal to] 1 and h [member of] [0,[l'.sub.n]].

Let us consider the maximum integer [k.sub.0] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [l'.sub.s] [less than or equal to] h for some s [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us assume now that [l'.sub.s] > h for every s [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. There exists some integer m with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By symmetry, without loss of generality we can assume that m [less than or equal to] [k.sub.0].

If m < [k.sub.0] and [l.sub.m] [less than or equal to] h, a similar argument gives the same bound for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, ([K.sup.0])' [less than or equal to] N [e.sup.N] [K.sup.0] + N and Theorem 4.2 implies (2).

(3) Assume first that N = 2; then {[l'.sub.n]} is the union of {[l.sup.1.sub.n]} and {[l.sup.2.sub.n]}. We denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the subsequence {[l.sup.i.sub.n]} in {[l'.sub.n]}, for i = 1, 2. Fix n [greater than or equal to] 1 and h [member of] [0,[l'.sub.n]]. By symmetry, without loss of generality we can assume that there exist [k.sub.1] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since the other case is similar.

If there is no k with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume now that there exists k with [n.sup.2.sub.k] [member of] ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

If there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If there exists [k.sub.3] verifying the next three conditions simultaneously:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(b) there exists [m.sub.3] [less than or equal to] [k.sub.3] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(c) for every k [member of] ([k.sub.3], [k.sub.2]] we have [([GAMMA].sup.0.sub.km]).sup.2](h) > [([K.sup.0]).sup.2] for every m [less than or equal to] k, then there exists [m.sub.0] > [k.sub.2] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]: In fact, seeking for a contradiction, let us assume that there exists [m.sub.0] [member of] ([k.sub.3] + 1 , [k.sub.2]] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If for any k with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we have ([[GAMMA].sup.0.sub.km]).sup.2](h) > (K.sup.0]).sup.2] for every m [less than or equal to] k, let us define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As in the last case, then there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, ([K.sup.0])' [less than or equal to] 2[([K.sup.0]).sup.1] + 2[([K.sup.0]).sup.2] and Theorem 4.2 implies (3) with N = 2. The result for N sequences is obtained by applying N [right arrow] 1 times this result for 2 sequences.

(4) is a direct consequence of (3) and Proposition 4.1.

(5) is a direct consequence of (3) and Theorem 4.3.

(6) Since {[x.sub.n]} [??] H, by Theorem 4.2 and Proposition 4.2, for each M > N there exist [k.sub.0] and h [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for every m [greater than or equal to] 1.

Consider m [greater than or equal to] 1. By symmetry, without loss of generality we can assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, ([K.sup.0])' [greater than or equal to] M - N - 1 for every M [greater than or equal to] N, and hence ([K.sup.0])' = [infinity]. Then {[l'.sub.n]} [??] H by Theorem 4.2.

(7) First, we want to remark the following elementary fact: If i [less than or equal to] j and [sigma](i) > [sigma](j), then [absolute value of i-j] < 2N: [absolute value of i-j] = j - i < j - [sigma](j) + [sigma](i) - i [less than or equal to] 2N.

Fix n [greater than or equal to] 1 and h [member of] [0,[l'.sub.n]]. There exists [sigma](m) with [[GAMMA].sup.0.sub.[sigma](n)[sigma](n)](h) [less than or equal to] [K.sup.0]. By symmetry, without loss of generality we can assume that [sigma](m) [less than or equal to] [sigma](n).

If m = n, then [sigma](m) = [sigma](n) and {[[GAMMA].sup.0.sub.nn])'(h) = [[GAMMA].sup.0.sub.[sigma](n)[sigma](n)] (h) [less than or equal to] [K.sup.0].

We consider now the case [sigma](m) < [sigma](n).

If m > n, then m - n < 2N.

If [B'.sub.n](h) > m, then [l'.sub.k] > h for every k [member of] (n, m] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We deal now with the case m < n. Notice first that [sigma]([m, n]) [subset] [m [right arrow] N, n + N] and [m + N, n [right arrow] N] [subset] [[sigma](m), [sigma](n)\; then, in [sigma]([m, n]) [[sigma](m), [sigma](n)\ there are at most 4N integers.

If [A'.sub.n](h) [greater than or equal to] m, then [l'.sub.k] > h for every k [member of] ([A'.sub.n](h), n), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Acknowledgement. The researches of Ana Portilla, Jose M. Rodriguez and Eva Touris were partially supported by three grants from M.E.C. (MTM 2006-11976, MTM 2006-13000-C03-02 and MTM 2007-30904-E), Spain.

The research of Eva Touris was partially supported by a grant from U.C.IIIM./C.A.M. (CCG08-UC3M/ESP-4516), Spain.

Received: May 27, 2009.

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St. Louis University (Madrid Campus)

Avenida del Valle 34, 28003 Madrid, Spain

E-mail address: aportil2@slu.edu

Universidad Carlos III de Madrid

Escuela Politecnica Superior

Departamento de Matematicas

Avenida de la Universidad 30, 28911 Leganes (Madrid), Spain

E-mail address: jomaro@math.uc3m.es

Universidad Carlos III de Madrid

Escuela Politecnica Superior

Departamento de Matematicas

Avenida de la Universidad 30, 28911 Leganes (Madrid), Spain

E-mail address: etouris@math.uc3m.es
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Author:Portilla, Ana; Rodriguez, Jose M.; Touris, Eva
Publication:Journal of Advanced Mathematical Studies
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Date:Jul 1, 2009
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