Moebius rigidity for compact deformations of negatively curved manifolds

Kingshook Biswas

paper

Moebius rigidity for compact deformations of negatively curved manifolds

arXiv:1812.04888

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 $i d : (X, g_{0}) \to (X, g_{1})$ 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 $i d^_{g_{0}, g_{1}} : \partial_{g_{0}} X \to \partial_{g_{1}} X$ . We show that if the boundary map $i d^_{g_{0}, g_{1}}$ is Moebius (i.e. preserves cross-ratios), then it extends to an isometry $F : (X, g_{0}) \to (X, g_{1})$ .