The local maxima of maximal injectivity radius among hyperbolic surfaces

TL;DR

This paper proves that within the Teichmüller space of hyperbolic surfaces, the function measuring the maximal injectivity radius has no local maxima other than the global maximum, indicating a certain global optimality property.

Contribution

The paper establishes a new global property of the injectivity radius function on Teichmüller space, showing the absence of non-global local maxima.

Findings

Maximal injectivity radius function has no local maxima except the global maximum.

The result applies to complete, orientable, finite-area hyperbolic surfaces.

Provides insight into the geometric structure of Teichmüller space.

Abstract

The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.