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

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

2007-02-25

arxiv.org/abs/math/0702750v1

Mathematics

Differential Geometry

12 pages

Scientific paper

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 1980's. 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.

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