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

By M. Makowski, Julian Scheuer. 2013

63 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/de3ff19d3266b3756aba08d22a8e20f8dbe454f2

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In Applied Mathematics

1. Mattia Fogagnolo
2. L. Mazzieri
3. Virginia Agostiniani
4. Zoltán M. Balogh
5. Alexandru Krist'aly
6. M. Rigoli

Citations in the corpus

• The Spherical Convex Floating Body (2014, unknown): “…of convex bodies in more general spaces, including the sphere, has seen tremendous progress over the last decades, see e.g. [1–3, 8, 25, 34, 63, 66, 74], and in recent years classical spherical convex geometry has made progress at a rapid pace, see [4] [9] [17] [20] [23] [24] [53] [56] [64] [80].”
• Minkowski Inequalities via Nonlinear Potential Theory (2019, background): “This approach, which is extremely clean and quite flexible, has found remarkable applications also in non Euclidean contexts (see [10, 28, 29, 30, 21, 47]).”