  In [dLMu05], DeLellis and M\"uller proved a quantitative version of Codazzi's
theorem, namely for a smooth embedded surface $\ \Sigma \subseteq \mathbb{R}^3\
$ with area normalized to $\ {\cal H}^2(\Sigma) = 4 \pi\ $, it was shown that
$\ \parallel A_\Sigma - id \parallel_{L^2(\Sigma)} \leq C \parallel A^0_\Sigma
\parallel_{L^2(\Sigma)}\ $, and building on this, closeness of $\ \Sigma\ $ to
a round sphere in $\ W^{2,2}\ $ was established, when $\ \parallel A^0_\Sigma
\parallel_{L^2(\Sigma)}\ $ is small. This was supplemented in [dLMu06] by
giving a conformal parametrization $\ S^2 \stackrel{\approx}{\longrightarrow}
\Sigma\ $ with small conformal factor in $\ L^\infty\ $, again when $\
\parallel A^0_\Sigma \parallel_{L^2(\Sigma)}\ $ is small. In this article, we
extend these results to arbitrary codimension. In contrast to [dLMu05], our
argument is not based on the equation of Mainardi-Codazzi, but instead uses the
monotonicity formula for varifolds.
