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The existence of embedded minimal hypersurfaces. (English) Zbl 1284.53057

The goal of the paper is to give a new proof of the result that was previously proven by J. T. Pitts [Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton, New Jersey: Princeton University Press; University of Tokyo Press (1981; Zbl 0462.58003)] for dimensions \(2\leq n\leq 5\) and by R. Schoen and L. Simon [Commun. Pure Appl. Math. 34, 741–757 (1981; Zbl 0497.49034)] in the general case: For a smooth closed \((n+1)\)-dimensional Riemannian manifold \(M\) there is a nontrivial embedded minimal hypersurface without boundary with a singular set of Hausdorff dimension at most \(n-7\).
The proof follows the general line of reasoning in two aforementioned works, but significantly simplifies some parts of it. The main idea of the proof is using families of sweepouts of \(M\). There is a natural notion of homotopy for sweepouts, and in the paper it is proven that for each homotopically closed family of sweepouts (such families are generated, for example, by Morse functions) there is a sequence of sets in this family that converges to a desired hypersurface. The proof of this fact is divided into “existence” and “regularity” parts, and the main improvement of the paper is contained in the former.
The authors introduce the notion of almost minimizing varifolds different from the definition used by Pitts, Schoen, and Simon. Roughly speaking, a varifold is almost minimizing if any deformation that decreases its volume must pass through a hypersurface with sufficiently big volume. This notion allows to prove the existence of a sequence of almost minimizing hypersurfaces in a family of sweepouts that converge to a stationary varifold. Then the authors prove that the sequence obtained can be modified by the so-called replacements so that the resulting varifold is induced by a minimal hypersurface of the desired type. In this “regularity” part the curvature estimates obtained by Schoen and Simon are used. So, in the paper the existing proof is modified using some new ideas that make it simpler and more accessible.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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