Measure-theoretic rigidity for Mumford curves

TL;DR

This paper extends measure-theoretic rigidity concepts from hyperbolic Riemann surfaces to Mumford curves, showing that boundary measure conditions can determine isomorphism of these nonarchimedean curves.

Contribution

It establishes a measure-theoretic criterion involving harmonic measures for isomorphism of Mumford curves, generalizing classical rigidity results to a nonarchimedean setting.

Findings

Absolute continuity of boundary maps implies isomorphism of special fibers.

Additional conditions on harmonic measures ensure full isomorphism of Mumford curves.

The proof combines rigidity theorems, boundary map construction, and classical algebraic geometry results.

Abstract

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain nonarchimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage-Enriques-Petri on equations for the canonical embedding of a curve.