Optimal regularity for minimizers of the prescribed mean curvature functional over isotopies

Abstract

We prove the optimal $C^{1,1}$ regularity for minimizers of the prescribed mean curvature functional over isotopy classes. As an application, we find an embedded sphere of prescribed mean curvature in the round 3-sphere for an open dense set of prescribing functions with $L^{\infty }$ norm at most 0.547.

70 pages, final version to appear in Camb. J. Math