In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many embedded minimal hypersurfaces with free boundary which are properly embedded.

中文翻译：

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自由边界最小超曲面的最小-最大理论，I：正则性理论

1960 年代，Almgren 发起了一个程序，使用最小-最大方法在紧凑流形中寻找最小超曲面。当流形没有边界时，Pitts 和 Schoen-Simon 在 1980 年代很大程度上推进了这个程序。在本文中，我们完成了具有非空边界的一般紧流形的程序。因此，我们证明了在任何紧致光滑欧几里得域中存在具有自由边界的光滑嵌入式最小超曲面。将我们的普遍存在结果结合 Marques 和 Neves 的工作应用表明，对于任何具有非负 Ricci 曲率和凸边界的紧黎曼流形，都存在无限多个具有适当嵌入的自由边界的嵌入最小超曲面。