Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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