On compact embedded Weingarten hypersurfaces in warped products

Abstract

We show that a compact embedded starshaped r-convex hypersurface of certain warped products satisfying Hr=aH+b with a⩾0, b>0, where H and Hr 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.