Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

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.