The topology of balls and Gromov hyperbolicity of Riemann surfaces

2010/09/20 by Pérez, Jesús Gonzalo, Portilla, Ana, Rodríguez, José M +1
#30F20 #32G15 #53C21 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.3881

Abstract

For each k > 0 we find an explicit function f_k such that the topology of S inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian surface (compact or noncompact) with K≥ -k², every point p on the surface, and every r. Using this result, we obtain a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a Riemann surface S* (with its own Poincaré metric) obtained by deleting from one original surface S any uniformly separated union of continua and isolated points.

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