Calculus of Variations and Geometric Measure Theory

C. De Lellis - S. Müller

A $C^0$--estimate for nearly umbilical surfaces.

created by delellis on 08 Mar 2005
modified on 03 May 2011


Published Paper

Inserted: 8 mar 2005
Last Updated: 3 may 2011

Journal: Calc. Var. Partial Differential Equations
Volume: 26
Number: 3
Pages: 283-296
Year: 2006


Let $\Sigma$ be a smooth compact connected surface of $*R*^3$ without boundary and denote by $A$ its second fundamental. In a previous paper we proved that, if $\
A- ({\rm tr } A/2) {\rm Id}\
_{L^2 (\Sigma)}$ is small, then $\Sigma$ is $W^{2,2}$--close to a round sphere. In this note we show that, in addition, the metric of $\Sigma$ is $C^0$--close to the standard metric of $*S*^2$.

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Keywords: rigidity, second fundamental form, umbilical surfaces