Rigidity of flat surfaces under the boundary measure

TL;DR

This paper proves that for a closed flat surface of genus at least 2, the boundary measure class of its universal cover uniquely identifies the surface, revealing a rigidity property linking boundary measures to surface geometry.

Contribution

It establishes a rigidity theorem showing the boundary measure class of the universal cover uniquely determines the flat surface.

Findings

Boundary measure class uniquely determines the flat surface.

Gromov boundary measures encode geometric information.

Rigidity holds for genus g ≥ 2 surfaces.

Abstract

Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.