# Moebius rigidity for simply connected, negatively curved surfaces

## TL;DR

This paper proves that Moebius homeomorphisms between the boundaries at infinity of simply connected, negatively curved surfaces extend to isometries, generalizing Otal's rigidity theorem without requiring cocompactness or group actions.

## Contribution

It establishes Moebius boundary maps extend to isometries for negatively curved surfaces without cocompactness assumptions, broadening Otal's rigidity results.

## Key findings

- Moebius boundary maps extend to isometries.
- No cocompactness or group action assumptions needed.
- Generalizes Otal's marked length spectrum rigidity theorem.

## Abstract

Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalization of Otal's marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal's theorem asserts that if $X, Y$ admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map $f$ is Moebius and equivariant with respect to these actions then it extends to an isometry. In our case there are no cocompactness or equivariance assumptions, indeed the isometry groups of $X, Y$ may be trivial.