Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

Abstract

Assume $h$ is a positive function on the unit three-sphere which satisfies the pinching condition $h<h_0\approx 0.547$. We prove the existence of at least two embedded two-spheres with prescribed mean curvature $h$. The same result holds for sign-changing functions $h$ satisfying $|h|<h_0$ under a mild assumption on the zero set.

15 pages, comments welcome!