## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Spacelike Graphs with Parallel Mean Curvature

Isabel M.C. Salavessa

#### Abstract

We consider spacelike graphs $\Gamma_f$ of simple products $(M\times N, g\times -h)$ where $(M,g)$ and $(N,h)$ are Riemannian manifolds and $f:M\rightarrow N$ is a smooth map. Under the condition of the Cheeger constant of $M$ to be zero and some condition on the second fundamental form at infinity, we conclude that if $\Gamma_f\subset M\times N$ has parallel mean curvature $H$ then $H=0$. This holds trivially if $M$ is closed. If $M$ is the $m$-hyperbolic space then for any constant $c$, we describe an explicit foliation of ${\mathbb H}^m\times \mathbb R$ by hypersurfaces with constant mean curvature $c$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 1 (2008), 65-76.

Dates
First available in Project Euclid: 22 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1203692447

Digital Object Identifier
doi:10.36045/bbms/1203692447

Mathematical Reviews number (MathSciNet)
MR2406087

Zentralblatt MATH identifier
1146.53036

#### Citation

Salavessa, Isabel M.C. Spacelike Graphs with Parallel Mean Curvature. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 65--76. doi:10.36045/bbms/1203692447. https://projecteuclid.org/euclid.bbms/1203692447