A quasiconformal Hopf soap bubble theorem

Abstract

We show that any compact surface of genus zero in $${\mathbb {R}}^3$$ R 3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon’s quasiconformal Bernstein theorem. The result generalizes, among others, Hopf’s theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.

Uniqueness of immersed spheres in three-manifolds

“…In [10], the first two authors showed the converse statement, i.e., that any C 2 ovaloid in R 3 that satisfies the Weingarten inequality must be totally umbilic along four disjoint great semicircles. For other uniqueness theorems of ovaloids, or immersed spheres in R 3 , satisfying the Weingarten inequality, see [8,11,12,13,20].…”

Constant mean curvature hypersurfaces in $\mathbb{H}^n\times\mathbb{R}$ with small planar boundary

“…Bryant [Br] studied closed SW-surfaces using complex analysis. More recently many new results and examples about Special Weingarten surfaces were produced by different authors [AlEsGa,BuOr,CaCa,CoFeTe,EsMe,FeGaMi,GaMaMi,GaMi1,GaMi2,GaMiTa,KuSt,SaTo1,SaTo2]. We point out that, in most of the cited works, W is expressed in terms of the mean curvature H and the Gaussian curvature K, resulting in the symmetry of W with respect to the principal curvatures.…”

On the Convergence of Non-Integer Linear Hopf Flow

“…The question of what values of umbilic slope are possible on various spheres is an area of active research [11] [5]. The derived Codazzi-Mainardi equation (2.3) is a necessary integrability condition for a C 3 -smooth surface to be rotationally symmetric and has some striking consequences in this direction, two of which are the following proposition and theorem.…”