On compact embbeded Weingarten hypersurfaces in warped products

TL;DR

This paper characterizes certain Weingarten hypersurfaces in warped products, showing they are slices under specific curvature conditions, and establishes stability and classification results in space forms and Euclidean space.

Contribution

It proves that compact starshaped r-convex hypersurfaces satisfying a linear relation between mean and r-th mean curvature are slices, and classifies such hypersurfaces in space forms without starshapedness.

Findings

Hypersurfaces satisfying H_r=aH+b are slices in warped products.

In space forms, such hypersurfaces are geodesic spheres without starshapedness.

Near-equality of Hr-aH-b implies the hypersurface is close to a geodesic sphere.

Abstract

We show that compact embedded starshaped $r$-convex hypersurfaces of certain warped products satisfying $H_r=aH+b$ with $a\geqslant 0$, $b>0$, where $H$ and $H_r$ are respectively the mean curvature and $r$-th mean curvature is a slice. In the case of space forms, we show that without the assumption of starshapedness, such Weingarten hypersurfaces are geodesic spheres. Finally, we prove that, in the case of space forms, if $Hr-aH-b$ is close to $0$ then the hypersurface is close to geodesic sphere for the Hausdorff distance. We also prove an anisotropic version of this stability result in the Euclidean space.