Local and infinitesimal rigidity of simply connected negatively curved manifols

TL;DR

This paper establishes local and infinitesimal rigidity results for negatively curved, simply connected manifolds, showing that boundary Moebius maps imply isometries under certain conditions.

Contribution

It proves that boundary Moebius maps determine the metric up to isometry for compactly supported deformations of negatively curved manifolds.

Findings

Boundary maps are Moebius if and only if the metrics are isometric.

Metrics close in the $C^{2,eta}$ topology with matching volume are isometric.

Rigidity holds for deformations fixed outside compact sets.

Abstract

Let $(X, g_0)$ be a simply connected, complete, negatively curved Riemannian manifold. We prove local and infinitesimal rigidity results for compactly supported deformations of the metric $g_0$. For any negatively curved metric $g$ equal to $g_0$ outside a compact, the identity map of $X$ induces a natural boundary map between the boundaries at infinity of $X$ with respect to $g_0$ and $g$. We show that if $(g_t)$ is a smooth 1-parameter family of negatively curved metrics all equal to $g_0$ outside a fixed compact then if all the boundary maps (between the boundaries of $X$ with respect to $g_0$ and $g_t$) are Moebius then the metrics $g_t$ are all isometric to $g_0$. We also show that given a compact $K$ in $X$, there is a neighbourhood of $g_0$ in the $C^{2,\alpha}$ topology such that for any negatively curved metric $g$ in this neighbourhood which is equal to $g_0$ outside $K$, if the boundary map is Moebius and the $g_0$ and $g$ volumes of $K$ agree then $g$ is isometric to $g_0$.