Control of the isoperimetric deficit by the Willmore deficit

Abstract

In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit. Let us consider the classM of smoothly embedded surfaces Σ ⊂ R of sphere type with enclosed inner region ΩΣ ⊂ R, and let us denote by ar(Σ) and vol(ΩΣ) the twodimensional Hausdorff measure of Σ and the three-dimensional Lebesgue measure of ΩΣ, respectively. For Σ ∈M we define the isoperimetric ratio I(Σ) := ar(Σ) vol(ΩΣ) 2 3 , (1) and the Willmore energy W(Σ) := 1 4 ∫ Σ |H| dH, (2) where H denotes the mean curvature of the surface Σ. Both functionals are invariant under dilations and translations; round spheres are in both cases the unique minimizers, in particular I(Σ) ≥ I(S) = (6 √ π) 2 3 , W(Σ) ≥ W(S) = 4π for all Σ ∈ M. We consider for any such Σ and both functionals the corresponding deficits, that is the difference from the optimal value. Our main result is the following control of the isoperimetric deficit by the Willmore deficit. Theorem 1. For all c0 > 0 there exists a universal constant C > 0 such that I(Σ)− I(S) ≤ C ( W(Σ)−W(S) ) (3) for all Σ ∈M with I(Σ)− I(S) ≤ c0. Umbilical surfaces Σ ∈ M are by a classical Theorem of Codazzi round spheres. Theorem 1 can be also been seen as a quantitative version of this statement: An equivalent formulation of (3) is that I(Σ)− I(S) ≤ C 1 4 ∫