Flexible hyperbolic cone metrics on the genus 2 surface

TL;DR

This paper investigates the classification of hyperbolic cone metrics on genus 2 surfaces, identifying the conditions under which these metrics are rigid or flexible, and quantifying the orbits of non-rigid metrics under the mapping class group.

Contribution

It demonstrates that there are exactly nine mapping class group orbits of non-rigid hyperbolic cone metrics on genus 2 surfaces, extending understanding of metric rigidity and flexibility.

Findings

Nine orbits of non-rigid metrics identified

Use of Erlandsson, Leininger, and Sadanand's theorem

Clarification of rigidity conditions for hyperbolic cone metrics

Abstract

A negatively curved hyperbolic cone metric on a surface is rigid if it is determined by the support of its Liouville current. We use a theorem of Erlandsson, Leininger, and Sadanand to show that there are nine mapping class group orbits of equivalence classes of non-rigid (aka flexible) metrics in the case of the genus 2 surface.