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

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.

arXiv comment32 pages; comments welcome. arXiv admin note: substantial text overlap with arXiv:1707.08012

References in corpus (3)

• Min-max theory for free boundary minimal hypersurfaces I - regularity theory
• Min-max theory for constant mean curvature hypersurfaces
• Entropy of closed surfaces and min-max theory

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• The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces
• Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume
• Multiplicity one for min-max theory in compact manifolds with boundary and its applications
• Generic Regularity of Minimal Hypersurfaces in Dimension 8
• Existence of curves with constant geodesic curvature in a Riemannian 2-sphere
• Min-max theory for networks of constant geodesic curvature