This article is concerned with the existence theory of closed minimal hypersurfaces in closed Riemannian manifolds of dimension at least three. These hypersurfaces are critical points for the area functional, and hence their study can be seen as a high-dimensional generalization of the classical theory of closed geodesics (Birkhoff, Morse, Lusternik, Schnirel’mann,…). The best result until very recently, due to Almgren ([2], 1965), Pitts ([37], 1981), and Schoen–Simon ([43], 1981), was the existence of at least one closed minimal hypersurface in every closed Riemannian manifold.I will discuss the methods I have developed with André Neves, for the past few years, to approach this problem through the variational point of view. These ideas have culminated with the discovery that minimal hypersurfaces in fact abound.

中文翻译：

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最小的表面丰富

本文关注的是尺寸至少为3的闭合黎曼流形中闭合最小超曲面的存在理论。这些超曲面是区域功能的关键点，因此，它们的研究可以看作是封闭测地线经典理论（Birkhoff，Morse，Lusternik，Schnirel'mann等）的高维概括。直到最近，由于Almgren（[2]，1965），Pitts（[37]，1981）和Schoen-Simon（[43]，1981），最好的结果是至少存在一个封闭的最小超曲面。我将讨论过去几年我与安德烈·内维斯（AndréNeves）开发的方法，以通过变化的观点来解决这个问题。这些想法以发现实际上存在大量超曲面的结果而达到顶峰。