## Illinois Journal of Mathematics

### Uniqueness of starshaped compact hypersurfaces with prescribed $m$-th mean curvature in hyperbolic space

#### Abstract

Let $\psi$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $\psi$, when evaluated on $M$, coincides with the $m$-th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980s. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigate the problem of uniqueness in hyperbolic space and show that uniqueness (up to a geometrically trivial transformation) holds under the same conditions under which existence was established.

#### Article information

Source
Illinois J. Math., Volume 51, Number 2 (2007), 571-582.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138430

Digital Object Identifier
doi:10.1215/ijm/1258138430

Mathematical Reviews number (MathSciNet)
MR2342675

Zentralblatt MATH identifier
1128.53007

#### Citation

Barbosa, João Lucas M.; de Lira, Jorge H. S.; Oliker, Vladimir. Uniqueness of starshaped compact hypersurfaces with prescribed $m$-th mean curvature in hyperbolic space. Illinois J. Math. 51 (2007), no. 2, 571--582. doi:10.1215/ijm/1258138430. https://projecteuclid.org/euclid.ijm/1258138430