Existence of hypersurfaces with prescribed mean curvature I – generic min-max
By Xin Zhou, Jonathan J. Zhu. Cambridge Journal of Mathematics, 2018
82 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Computational Mechanics, Surgery
View Paper: https://www.semanticscholar.org/paper/2d49fc8553ee401f9d0888f10be9d44aa1a35341
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In Applied Mathematics
In Computational Mechanics
In Surgery
Citations in the corpus
- Generalized soap bubbles and the topology of manifolds with positive scalar curvature (2020, methodology): “nd an extension of it) can also be proven using spinors and the index theorem, see [5]. 4The prescribed-mean-curvature functional has recently been considered in other geometric problems as well, cf. [32, 1, 31]. SOAP BUBBLES AND POSITIVE SCALAR CURVATURE 5 curvature might be dramatically different n”
- On the Multiplicity One Conjecture in min-max theory (2019, unknown): “Here we present an adaption to multi-parameter families of the min-max theory for hypersurfaces with prescribed mean curvature (abbreviated as PMC) established by the author with Zhu [51, 52].”