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MiS Preprint
47/2003

Sharp rigidity estimates for nearly umbilical surfaces

Camillo De Lellis and Stefan Müller

Abstract

A classical theorem in differential geometry states that if $\Sigma\subset {\bf R}^3$ is a compact connected surface wthout boundary and all points of $\Sigma$ are umbilical, then $\Sigma$ is a a round sphere and therefore its second fundamental form $A$ is a constant multiple of the identity. In this paper we give a sharp quantitative version of this theorem. More precisely we show that if the $L^2$ norm of the traceless part of $A$ is small, then $A$ is $L^2$ near to a constant multiple of the identity.

Received:
May 22, 2003
Published:
May 22, 2003
MSC Codes:
53A05, 53C24, 58J90
Keywords:
rigidity, umbilical surfaces, second fundamental form

Related publications

inJournal
2006 Repository Open Access
Camillo De Lellis and Stefan Müller

A C-0 estimate for nearly umbilical surfaces

In: Calculus of variations and partial differential equations, 26 (2006) 3, pp. 283-296