# Moebius rigidity for compact deformations of negatively curved manifolds

## TL;DR

This paper proves that for certain negatively curved manifolds, a boundary map that preserves cross-ratios implies the existence of a corresponding isometry between the manifolds, extending Moebius rigidity results.

## Contribution

It establishes a Moebius rigidity theorem for compact deformations of negatively curved manifolds, showing boundary cross-ratio preservation implies isometry.

## Key findings

- Boundary Moebius maps extend to isometries.
- Rigidity holds under specified curvature bounds.
- Extension applies to compactly deformed negatively curved manifolds.

## Abstract

Let $(X, g_0)$ be a complete, simply connected Riemannian manifold with sectional curvatures $K_{g_0}$ satisfying $-b^2 \leq K_{g_0} \leq -1$ for some $b \geq 1$. Let $g_1$ be a Riemannian metric on $X$ such that $g_1 = g_0$ outside a compact in $X$, and with sectional curvatures $K_{g_1}$ satisfying $K_{g_1} \leq -1$. The identity map $id : (X, g_0) \to (X, g_1)$ is bi-Lipschitz, and hence induces a homeomorphism between the boundaries at infinity of $(X, g_0)$ and $(X, g_1)$, which we denote by $\hat{id}_{g_0, g_1} : \partial_{g_0} X \to \partial_{g_1} X$. We show that if the boundary map $\hat{id}_{g_0, g_1}$ is Moebius (i.e. preserves cross-ratios), then it extends to an isometry $F : (X, g_0) \to (X, g_1)$.