*Published Paper*

**Inserted:** 22 may 2003

**Last Updated:** 3 may 2011

**Journal:** Journal of Differential Geometry

**Volume:** 69

**Number:** 75--110

**Year:** 2003

**Abstract:**

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.

For the most updated version and eventual errata see the page

http:/www.math.uzh.ch*index.php?id=publikationen&key1=493
*

**Keywords:**
rigidity, second fundamental form, umbilical surfaces