CMC hypersurfaces on riemannian and semi-riemannian manifolds

TL;DR

This paper constructs explicit examples of constant mean curvature hypersurfaces in semi-Riemannian manifolds, including de Sitter, anti de Sitter, and Minkowski spaces, expanding understanding of their geometric properties.

Contribution

It provides new explicit families of CMC hypersurfaces in various semi-Riemannian manifolds, including complete immersions and classifications of extendable closed hypersurfaces.

Findings

Realization of every h in [-1, -2√(n-1)/n) as curvature of a complete immersion in de Sitter space.

Three types of CMC immersions in Minkowski space.

Five types of CMC immersions in de Sitter and anti de Sitter spaces.

Abstract

In this paper we show explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-riemannian manifolds with constant sectional curvature. In particular, we prove that every h in [-1,-2 sqrt{n-1}/n) can be realized as the constant curvature of a complete immersion of S_1^{n-1} x R in the (n+1)-dimensional de Sitter space S_1^{n+1}. We provide 3 types of immersions with cmc in the Minkowski space, 5 types of immersion with cmc in the de Sitter space and 5 types of immersion with cmc in the anti de Sitter space. At the end of the paper we analyze the families of examples that can be extended to closed hypersurfaces.