Abstract
Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with nonpositive sectional curvature for $n = 2,3$ and $4$. We prove the following Faber-Krahn type inequality for the first eigenvalue $\lambda _{1}$ of the mixed boundary problem. A domain $\Omega$ outside a closed convex subset $C$ in $M$ satisfies \[ \lambda _{1}(\Omega )\geq \lambda _{1}(\Omega ^{*}) \] with equality if and only if $\Omega$ is isometric to the half ball $\Omega$ in $\mathbb{R}_{n}$, whose volume is equal to that of $\Omega$. We also prove the Sobolev type inequality outside a closed convex set $C$ in $M$.
Citation
Keomkyo Seo. "Geometric inequalites outside a convex set in a Riemannian manifold." J. Math. Kyoto Univ. 47 (3) 657 - 664, 2007. https://doi.org/10.1215/kjm/1250281030
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