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

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