Diameter of Compact Riemann Surfaces

TL;DR

This paper establishes the exact diameter of generalized Bolza surfaces of any genus greater than one, linking it to the radii of their fundamental polygons, thus providing the first precise measurement for a class of compact hyperbolic surfaces.

Contribution

It proves that the diameter of generalized Bolza surfaces equals the radii of their fundamental polygons, offering the first exact diameter result for certain compact hyperbolic manifolds.

Findings

Diameter of generalized Bolza surfaces equals the radii of their fundamental polygons

First exact diameter result for a class of compact hyperbolic surfaces

Provides new insights into geometric properties of high-genus 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.