ON COMPACT EMBBEDED WEINGARTEN HYPERSURFACES IN WARPED PRODUCTS

Julien Roth (1, 2) , Abhitosh Upadhyay (3)

1 Université Gustave Eiffel
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
3 IIT GOA - INDIAN INSTITUTE OF TECHNOLOGY GOA

We show that compact embedded starshaped r-convex hypersurfaces 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.

Domaines: Géométrie différentielle [math.DG]

Julien Roth, Abhitosh Upadhyay. ON COMPACT EMBBEDED WEINGARTEN HYPERSURFACES IN WARPED PRODUCTS. Journal of Mathematical Analysis and Applications, 2022, 517 (1), pp.126593. ⟨hal-03682396⟩