Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

AbstractIn this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface

A characterization of Gromov hyperbolicity of surfaces with variable negative curvature

In this paper we show that, in order to check Gromov hyperbolicity of any surface with curvature K≤ −k² < 0, we just need to verify the Rips condition on a very small class of triangles, namely, those contained in simple closed geodesics. This result is, in fact, a new characterization of Gromov hyperbolicity for this kind of surfaces

The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces

27 pages, no figures.-- MSC2000 codes: 30F20, 30F45.MR#: MR2243795 (2007e:30063)Zbl#: Zbl 1108.30031We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface S* obtained by deleting a closed set from one original surface S. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.Research by all three authors partially supported by a grant from DGI (BFM 2003-04870), Spain. In addition, research by third author (Eva Tourís) was partially supported by a grant from DGI (BFM 2000-0022), Spain.Publicad

A real variable characterization of Gromov hyperbolicity of flute surfaces

23 pages, 1 figure.-- MSC2000 codes: 41A10, 46E35, 46G10.-- ArXiv pre-print available at: http://arxiv.org/abs/0806.0093Previously presented as Communication at International Congress of Mathematicians 2006 (ICM2006, Madrid, Spain, Aug 22-30, 2006).Preaccepted for publication at: Osaka Journal of MathematicsIn this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric.Research partially supported by three grants from M.E.C. (MTM 2006-11976, MTM 2006-13000-C03-02 and MTM 2007-30904-E), Spain.No publicad

Gromov hyperbolicity of Riemann surfaces

20 pages, no figures.-- MSC2000 codes: 30F, 30F20, 30F45.MR#: MR2286916 (2007k:30080)Zbl#: Zbl 1115.30050In this paper we study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its "building block components". We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.The first author’s research is partially supported by a grant from DGI (BFM 2003-04870), Spain. The second author’s research is partially supported by a grant from DGI (BFM 2000-0022), Spain.Publicad

Gromov hyperbolicity through decomposition of metric spaces

32 pages, no figures.-- MSC2000 codes: 30F20, 30F45.-- A complementary work to this paper was published in: The Journal of Geometric Analysis 14(1): 123-149 (2004), http://hdl.handle.net/10016/6464MR#: MR2047877 (2005a:30076)Zbl#: Zbl 1051.30036We study the hyperbolicity of metric spaces in the Gromov sense. We deduce the hyperbolicity of a space from the hyperbolicity of its “building block components”. These results are valuable since they simplify notably the topology of the space and allow to obtain global results from local information. We also study how the punctures and the decomposition of a Riemann surface in Y-pieces and funnels affect the hyperbolicity of the surface.First author partially supported by a grant from DGI (BMF 2000-0022) Spain. Second author supported by a grant from DGI (BMF 2000-0022) Spain.Publicad

The multiplication operator, zero location and asymptotic for non-diagonal Sobolev norms

14 pages, no figures.-- Online version published Jul 9, 2009.Article in press.In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which the product of the function and its derivative appears. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.Supported in part by three grants from M.E.C. (MTM 2006-13000-C03-02, MTM 2006-11976 and MTM 2007-30904-E) and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain.Publicad

The topology of balls and Gromov hyperbolicity of Riemann surfaces

19 pages, no figures.-- MSC2000 codes: 30F20, 30F45, 53C23.MR#: MR2091367 (2005e:53057)Zbl#: Zbl 1070.30019We prove that every ball in any non-exceptional Riemann surface with radius less or equal than $\frac 1 2\log 3$ is either simply or doubly connected. We use this theorem in order to study the hyperbolicity in the Gromov sense of Riemann surfaces. The results clarify the role of punctures and funnels of a Riemann surface in its hyperbolicity.Research by first two authors (A.P. and J.M.R.) was partially supported by a grant from DGI (BFM 2000-0022), Spain. Research by third author (E.T.)was supported by a grant from DGI (BFM 2000-0022), Spain.Publicad