CMC hypersurfaces of semi-Riemannian groups

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.