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QUANTITATIVE OSCILLATION ESTIMATES FOR ALMOST-UMBILICAL CLOSED HYPERSURFACES IN EUCLIDEAN SPACE

Part of: Calculus on manifolds; nonlinear operators Global differential geometry

Published online by Cambridge University Press:  17 April 2015

JULIAN SCHEUER
JULIAN SCHEUER*
Affiliation:
Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany email [email protected]
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Abstract

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We prove ${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is ${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of ${\it\delta}$ on ${\it\epsilon}$ within the class of uniformly convex hypersurfaces with bounded volume.

Keywords

pinchingalmost-umbilicalconvex hypersurface

MSC classification

Primary: 53C20: Global Riemannian geometry, including pinching
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C24: Rigidity results 58C40: Spectral theory; eigenvalue problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 133 - 144
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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