## Journal of Mathematics of Kyoto University

### Geometric inequalites outside a convex set in a Riemannian manifold

Keomkyo Seo

#### 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 $M$.

#### Article information

Source
J. Math. Kyoto Univ., Volume 47, Number 3 (2007), 657-664.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250281030

Digital Object Identifier
doi:10.1215/kjm/1250281030

Mathematical Reviews number (MathSciNet)
MR2402521

Zentralblatt MATH identifier
1144.58017

#### Citation

Seo, Keomkyo. Geometric inequalites outside a convex set in a Riemannian manifold. J. Math. Kyoto Univ. 47 (2007), no. 3, 657--664. doi:10.1215/kjm/1250281030. https://projecteuclid.org/euclid.kjm/1250281030