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 expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold.

Computing distances on Riemann surfaces

“…The diameter D g of S g is known [28], and therefore we can substitute the actual diameter, instead of the upper bound in equation ( 13), into equation ( 14) for k ⋆ to obtain…”