Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere

Abstract

We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex $C^2$-hypersurfaces. We apply these results to prove $C^{1,\beta }$-convergence of inverse F-curvature flows in the sphere to an equator in \mathbb{S}^{n+1} for embedded, closed, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.

23 pages. Compared to version 2, we removed the statement about the isoperimetric inequality since it turned out to be false. Several typos were corrected

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• The inverse mean curvature flow in warped cylinders of non-positive radial curvature
• The inverse mean curvature flow perpendicular to the sphere
• New pinching estimates for Inverse curvature flows in space forms
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• Weighted geometric inequalities for hypersurfaces in sub-static manifolds
• Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary
• Surfaces expanding by non-concave curvature functions
• Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball
• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
• Inverse curvature flows in Riemannian warped products
• Geometric inequalities involving three quantities in warped product manifolds
• On an inverse curvature flow in two-dimensional space forms
• Minkowski inequalities and constrained inverse curvature flows in warped spaces
• Blaschke-Santaló type inequalities and quermassintegral inequalities in space forms
• A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space
• Alexandrov-Fenchel type inequalities with convex weight in space forms
• A spectral isoperimetric inequality on the n-sphere for the Robin-Laplacian with negative boundary parameter