Cylinders in Riemann surfaces

Abstract

Suppose S is a compact Riemann surface, with a metric of constant curvature 1 . Let 3' be a simple closed geodesic on S of length L~. Then for small d, the family of geodesic segments of length 2d, perpendicular to and centered on % sweeps out a region homeomorphic to a cylinder. As d is enlarged, it must eventually happen that two endpoints from this family of segments intersect. Let do> 0 be the smallest value of d for which this occurs. Setting d = do, and removing the envelope of the endpoints, we shall refer to the resulting set as the cylinder about 3/, and denote it by C~. Note that C v is topologically an open cylinder, and has the property that any curve emanating from 3' and of length less than do must lie wholly within Cv. Now it can be shown (cf. [3]) that given c > 0, there exists c ' > 0 such that Lv c ' . The techniques in [3] involve fine points in the theory of canonical fundamental domains and moduli for certain types of discontinuous groups, as well as substantial computation. Because of the interest of the result, it seems worthwhile to present an alternative, somewhat more geometric approach to the theorem, which at the same time leads to an improved result.