A quasiconformal Hopf soap bubble theorem

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