Inserted: 22 may 2003
Last Updated: 3 may 2011
Journal: Journal of Differential Geometry
A classical theorem in differential geometry states that if $\Sigma\subset *R*^3$ is a compact connected surface without boundary and all points of $\Sigma$ are umbilical, then $\Sigma$ is 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 prove 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.
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Keywords: rigidity, second fundamental form, umbilical surfaces