A quasiconformal Hopf soap bubble theorem

TL;DR

This paper proves that compact genus-zero surfaces in Euclidean 3-space satisfying a quasiconformal curvature inequality are round spheres, solving an old problem and extending classical theorems in differential geometry.

Contribution

It establishes a spherical version of Simon's quasiconformal Bernstein theorem, generalizing Hopf's theorem and related classification results for elliptic Weingarten surfaces.

Findings

Any genus-zero surface with quasiconformal curvature inequality is a sphere

The result extends classical theorems for constant mean curvature and elliptic Weingarten surfaces

The proof uses the Bers-Nirenberg representation for elliptic equations with discontinuous coefficients

Abstract

We show that any compact surface of genus zero in Euclidean 3-space 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.