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29 March 2024
 
  » arxiv » 1812.4888

 Article overview


Moebius rigidity for compact deformations of negatively curved manifolds
Kingshook Biswas ;
Date 12 Dec 2018
AbstractLet $(X, g_0)$ 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 $id : (X, g_0) o (X, g_1)$ 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 $hat{id}_{g_0, g_1} : partial_{g_0} X o partial_{g_1} X$. 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 $F : (X, g_0) o (X, g_1)$.
Source arXiv, 1812.4888
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