*Published Paper*

**Inserted:** 8 mar 2005

**Last Updated:** 3 may 2011

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 26

**Number:** 3

**Pages:** 283-296

**Year:** 2006

**Abstract:**

Let $\Sigma$ be a smooth
compact connected surface of $*R*^3$ without
boundary and denote by $A$ its second fundamental.
In a previous paper we proved that, if $\

A- ({\rm tr } A/2) {\rm Id}\

_{L^2 (\Sigma)}$ is small, then
$\Sigma$ is $W^{2,2}$--close to a round sphere.
In this note we show that, in addition, the metric of $\Sigma$ is
$C^0$--close to the standard metric of $*S*^2$.

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