The topology of balls and Gromov hyperbolicity of Riemann surfaces

TL;DR

This paper establishes explicit bounds on the topology of metric balls in Riemann surfaces with curvature constraints and characterizes their Gromov hyperbolicity after certain deletions, aiding understanding of their geometric structure.

Contribution

It provides explicit functions bounding the topology of metric balls and offers a practical criterion for Gromov hyperbolicity of modified Riemann surfaces.

Findings

Explicit functions bounding topology of metric balls

Characterization of Gromov hyperbolicity after deletions

Applicable to surfaces with curvature ≥ -k^2

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\geq -k^2, 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 Poincar\'e metric) obtained by deleting from one original surface S any uniformly separated union of continua and isolated points.