Diameter of Compact Riemann Surfaces

Abstract

Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple closed-form expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameter of the maximally symmetric Riemann surface of any genus greater than 1 is equal to the radius of its fundamental polygon.