Asian Journal of Mathematics

Volume 20 (2016)

Number 5

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

Pages: 869 – 892

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n5.a2

Authors

Matthias Makowski

Julian Scheuer (Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg, Germany)

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 and 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 and 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 some Alexandrov–Fenchel type inequalities.

Keywords

rigidity, inverse curvature flows, sphere, Alexandrov–Fenchel inequality

2010 Mathematics Subject Classification

35J60, 53C21, 53C24, 53C44, 58J05

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Published 22 February 2017