CMC hypersurfaces with two principal curvatures

TL;DR

This paper classifies all hypersurfaces with constant mean curvature and exactly two principal curvatures in space forms with semi-Riemannian metrics, providing explicit immersions and parameter ranges.

Contribution

It offers a complete construction and explicit parametrizations of all such hypersurfaces, extending previous results to semi-Riemannian contexts.

Findings

Explicit immersions for all hypersurfaces with two principal curvatures

Parameter ranges for complete cmc hypersurfaces in various space forms

Optimal bounds for the traceless second fundamental form

Abstract

This paper explains the construction of all hypersurfaces with constant mean curvature -- cmc -- and exactly two principal curvatures on any space form endowed with a semi-riemannian metric. Here we will consider riemannian hypersurfaces as well as hypersurfaces with semi-riemannian metrics induced by the ambient space. We show explicit immersions for all these hypersurfaces. We will see that for any given ambient space, the family of cmc hypersurfaces with two principal curvatures depends on two parameters, $H$ and $C$. We describe the range for all possible $(H,C)$ associated with complete cmc hypersurfaces. We end the paper by finding optimal bounds for the traceless second fundamental form in terms of $H$ and $n$.