Existence of hypersurfaces with prescribed mean curvature I – generic min-max
Existence of hypersurfaces with prescribed mean curvature I – generic min-max
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DOI：
10.4310/cjm.2020.v8.n2.a2
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发表时间：
2018-08
影响因子：
1.6
通讯作者：
Xin Zhou;Jonathan J. Zhu
中科院分区：
文献类型：
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作者：
Xin Zhou;Jonathan J. Zhu
证明了对于封闭环境流形上的光滑规定性函数$h$，总是存在一个规定平均曲率$h$的非平凡、光滑、封闭超曲面。其解要么是一个具有整数多重性的内嵌极小超曲面，要么是一个具有1多重性的非极小几乎内嵌超曲面。更准确地说，我们证明了先前针对常平均曲率超曲面所建立的最小-极大理论，可以推广到对于某些类型的规定函数，包括光滑莫尔斯函数和非零解析函数，构造最小-极大规定平均曲率超曲面。特别地，我们不需要假设$h$有符号。
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.