## Journal of Mathematics of Kyoto University

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

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

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/kjm/1250281030

**Mathematical Reviews number (MathSciNet)**

MR2402521

**Zentralblatt MATH identifier**

1144.58017

**Subjects**

Primary: 53Cxx: Global differential geometry [See also 51H25, 58-XX; for related bundle theory, see 55Rxx, 57Rxx]

Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds

#### 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