Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ Σ\subseteq \mathbb{R}^3\$ with area normalized to $\ {\cal H}^2(Σ) = 4 π$, it was shown that $\ \parallel A_Σ- id \parallel_{L^2(Σ)} \leq C \parallel A^0_Σ\parallel_{L^2(Σ)}\$, and building on this, closeness of $\ Σ$ to a round sphere in $\ W^{2,2}\$ was established, when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\$ is small. This was supplemented in [dLMu06] by giving a conformal parametrization $\ S^2 \stackrel{\approx}{\longrightarrow} Σ$ with small conformal factor in $\ L^\infty\$, again when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\$ 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.

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