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Local and infinitesimal rigidity of simply connected negatively curved manifols | Kingshook Biswas
; | Date: |
17 Jul 2015 | 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
$g_0$. | Source: | arXiv, 1507.4974 | Services: | Forum | Review | PDF | Favorites |
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