Open Access
Summer 2007 Uniqueness of starshaped compact hypersurfaces with prescribed $m$-th mean curvature in hyperbolic space
João Lucas M. Barbosa, Vladimir Oliker, Jorge H. S. de Lira
Illinois J. Math. 51(2): 571-582 (Summer 2007). DOI: 10.1215/ijm/1258138430

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.

Citation

Download Citation

João Lucas M. Barbosa. Vladimir Oliker. Jorge H. S. de Lira. "Uniqueness of starshaped compact hypersurfaces with prescribed $m$-th mean curvature in hyperbolic space." Illinois J. Math. 51 (2) 571 - 582, Summer 2007. https://doi.org/10.1215/ijm/1258138430

Information

Published: Summer 2007
First available in Project Euclid: 13 November 2009

zbMATH: 1128.53007
MathSciNet: MR2342675
Digital Object Identifier: 10.1215/ijm/1258138430

Subjects:
Primary: 53C21
Secondary: 35J60

Rights: Copyright © 2007 University of Illinois at Urbana-Champaign

Vol.51 • No. 2 • Summer 2007
Back to Top