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Jan 2005 Optimal rigidity estimates for nearly umbilical surfaces
Camillo De Lellis, Stefan Müller
J. Differential Geom. 69(1): 075-110 (Jan 2005). DOI: 10.4310/jdg/1121540340

Abstract

Let $\Sigma \in \mathbf{R}^3$ be a smooth compact connected surface without boundary and denote by $A$ its second fundamental form. We prove the existence of a universal constant $C$ such that \begin{equation} \inf_{\lambda\in {\bf R}}\Vert A - \lambda \rm{Id} \Vert_{L^2(\Sigma)} \leq C \Vert A - \frac{\rm{tr}A}{2} \rm {Id} \Vert_{L^2(\Sigma)^\cdot} \end{equation} Building on this, we also show that, if the right-hand side of (1) is smaller than a geometric constant, Σ is W2,2–close to a round sphere.

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Camillo De Lellis. Stefan Müller. "Optimal rigidity estimates for nearly umbilical surfaces." J. Differential Geom. 69 (1) 075 - 110, Jan 2005. https://doi.org/10.4310/jdg/1121540340

Information

Published: Jan 2005
First available in Project Euclid: 16 July 2005

zbMATH: 1087.53004
MathSciNet: MR2169583
Digital Object Identifier: 10.4310/jdg/1121540340

Rights: Copyright © 2005 Lehigh University

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Vol.69 • No. 1 • Jan 2005
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