CMC hypersurfaces of semi-Riemannian groups

TL;DR

This paper investigates constant mean curvature hypersurfaces in semi-Riemannian Lie groups, revealing their geometric structure, stability, and classification under various curvature and transversality conditions.

Contribution

It extends the classification and geometric analysis of CMC hypersurfaces in semi-Riemannian groups, including new results on their structure, stability, and nullity properties.

Findings

CMC hypersurfaces are lateral classes of Lie subgroups under certain conditions

Complete minimal hypersurfaces are either transversal or have degenerate Gauss map

Complete hypersurfaces with large mean curvature are totally umbilical under bounded curvature conditions

Abstract

In this paper, we study the geometry of a connected oriented cmc Riemannian hypersurface $M$ of a semi-Riemannian group $G$ of Lie algebra $\mathfrak g$ and index 0 or 1. If $G$ is Riemannian and $M$ is compact and transversal to an element of $\mathfrak g$, we show that it is a lateral class of a closed embedded Lie subgroup of $G$; we also do this if $G$ is Lorentzian, provided $M$ has sufficiently large mean curvature. If $G$ is Riemannian semisimple and $M$ is compact, we prove that $M$ has degenerate Gauss map and minimal relative nullity at least 1. We also extend the above results to the case where $M$ is complete and noncompact. For a Riemannian $G$, we show that a minimal $M$ is either transversal to an element of $\mathfrak g$, hence stable, or has degenerate Gauss map and minimal relative nullity at least 1; for $M$ cmc and transversal to an element of $\mathfrak g$, if we ask the immersion to be proper and have bounded second fundamental form, then $M$ is also a lateral class of a closed embedded Lie subgroup of $G$, provided a certain growing condition on the size of the corresponding Gauss map is satisfied. Finally, for a Lorentzian group $G$, with sectional curvatures bounded from above on Lorentzian planes, we extend a result of Y. Xin, proving that a complete $M$ is totally umbilical, provided it is transversal to a timelike element of $\mathfrak g$, has large enough mean curvature and bounded hyperbolic Gauss map.