Lengthening deformations of singular hyperbolic tori

TL;DR

This paper characterizes infinitesimal deformations of singular hyperbolic tori that uniformly lengthen or shorten all closed geodesics, and examines the degeneration of these deformations as the surface approaches Euclidean geometry.

Contribution

It provides a detailed description of lengthening deformations for singular hyperbolic tori and analyzes their behavior in the Euclidean limit.

Findings

Characterization of lengthening/shrinking deformations of singular hyperbolic tori.

Analysis of degeneration behavior as the surface becomes Euclidean.

Insights into the geometric structure of deformations near the Euclidean limit.

Abstract

Let $S$ be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of $S$ lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when $S$ becomes Euclidean, i.e. very small.