Sharp rigidity estimates for nearly umbilical surfaces

Camillo De Lellis and Stefan Müller

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Submission date: 22. May. 2003
Pages: 29
published in: Calculus of variations and partial differential equations, 26 (2006) 3, p. 283-296 
DOI number (of the published article): 10.1007/s00526-006-0005-5
Bibtex
with the following different title: A C-0 estimate for nearly umbilical surfaces
MSC-Numbers: 53A05, 53C24, 58J90
Keywords and phrases: rigidity, umbilical surfaces, second fundamental form
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Abstract:
A classical theorem in differential geometry states that if is a compact connected surface wthout boundary and all points of are umbilical, then 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 norm of the traceless part of A is small, then A is near to a constant multiple of the identity.