2013/10/18 by Lamm, Tobias, Schätzle, Reiner M. · 1 citation
#49Q15 #53A05 #53A30 #53C21 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1310.4971
In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface Σ⊆ ℝ3 with area normalized to \cal H2(Σ) = 4 π , it was shown that ∥ AΣ- id ∥L2(Σ) ≤ C ∥ A0Σ∥L2(Σ) , and building on this, closeness of Σ to a round sphere in W2,2 was established, when ∥ A0Σ∥L2(Σ) is small. This was supplemented in [dLMu06] by giving a conformal parametrization S2 \stackrel≈\longrightarrow Σ with small conformal factor in L^∞ , again when ∥ A0Σ∥L2(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.