Existence of hypersurfaces with prescribed mean curvature I - Generic min-max

Xin Zhou; Jonathan J. Zhu

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arXiv · 1808.03527

Existence of hypersurfaces with prescribed mean curvature I - Generic min-max

Xin Zhou·Jonathan J. Zhu

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Abstract

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 smooth Morse functions and nonzero analytic functions. In particular we do not need to assume that $h$ has a sign.

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Xin Zhou, Jonathan J. Zhu. 2018-08-09. Existence of hypersurfaces with prescribed mean curvature I - Generic min-max. https://arxiv.org/abs/1808.03527

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