Local and infinitesimal rigidity of simply connected negatively curved manifols

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 $g_0$.