Existence of hypersurfaces with prescribed mean curvature I – generic min-max

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one.

More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that $h$ has a sign.

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Xin Zhou was supported by NSF grant DMS-1811293 and an Alfred P. Sloan Research Fellowship.

Jonathan J. Zhu was supported by NSF grant DMS-1607871.

Received 22 March 2019

Published 21 April 2020