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De Lellis and Müller (J Differ Geom 69:75–110, 2005) proved a quantitative version of Codazzi’s theorem, namely for a smooth embedded surface $${\Sigma \subseteq \mathbb{R}^3 }$$Σ⊆R3 with area normalized to $${\mathcal{H}^2(\Sigma) = 4 \pi}$$H2(Σ)=4π, it was shown that $${\parallel A_\Sigma - id \parallel_{L^2(\Sigma)} \leq C \parallel A^0_\Sigma \parallel_{L^2(\Sigma)}}$$‖AΣ-id‖L2(Σ)≤C‖AΣ0‖L2(Σ), and building on this, closeness of $${ \Sigma}$$Σ to a round sphere in $${ W^{2, 2}}$$W2,2 was established, when $${\parallel A^0_\Sigma \parallel_{L^2(\Sigma)} }$$‖AΣ0‖L2(Σ) is small. This was supplemented in De Lellis and Müller (Calc Var Partial Differ Equ 26(3):288–296, 2006) by giving a conformal parametrization $${ S^2 \stackrel{\approx}{\longrightarrow} \Sigma}$$S2⟶≈Σ with small conformal factor in $${ L^\infty}$$L∞, again when $${ \parallel A^0_\Sigma \parallel_{L^2(\Sigma)}}$$‖AΣ0‖L2(Σ) is small. In this article, we extend these results to arbitrary codimension. In contrast to De Lellis and Müller (J Differ Geom 69:75–110, 2005), our argument is not based on the equation of Mainardi–Codazzi, but instead uses the monotonicity formula for varifolds.

Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension