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.

中文翻译：

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具有规定平均曲率I的超曲面的存在–一般最小-最大

我们证明，对于封闭的环境歧管上的一组通用的光滑处方函数$ h $，始终存在具有规定平均曲率$ h $的非平凡，光滑，闭合的超曲面。解决方案要么是具有整数多重性的嵌入式最小超曲面，要么是多重性为1的非最小几乎嵌入式超曲面。更准确地说，我们证明了我们为恒定平均曲率超曲面开发的先前的min-max理论可以扩展为构造某些类别的处方函数（包括一组光滑函数以及所有非零值）的min-max指定平均曲率超曲面分析功能。特别是，我们不需要假设$ h $有一个符号。