Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

TL;DR

This paper proves the existence of at least two embedded spheres with prescribed mean curvature on the three-sphere, under certain pinching conditions on the curvature function, extending to sign-changing functions.

Contribution

It establishes the existence of multiple embedded spheres with prescribed mean curvature under new pinching conditions, including sign-changing functions.

Findings

Existence of at least two embedded spheres with prescribed mean curvature for certain functions.

Extension of results to sign-changing curvature functions under mild zero set assumptions.

The pinching condition h < h_0 approximately 0.547 is critical for the proofs.

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 $\vert h\vert < h_0$ under a mild assumption on the zero set.