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The structure of stable minimal hypersurfaces in \(\mathbb{R}^{n+1}\). (English) Zbl 0906.53004

The authors prove that, for \(n\geq 3,\) any complete orientable stable minimal hypersurface \(M\) in \({\mathbb R}^{n+1}\) cannot have more than one end. The proof is rather elegant. A harmonic function on \(M\) is constructed by an exhaustion procedure in which each compact sub-problem imposes Dirichlet boundary data equal to 1 in the portion of the boundary approaching one end and vanishing Dirichlet data in the portion approaching the other end (or ends). By applying the Sobolev inequality of J. H. Michael and L. M. Simon [Commun. Pure. Appl. Math. 26, 361-379 (1973; Zbl 0256.53006)] to exploit the dichotomy between the boundary values imposed in the two or more ends, the authors are able to show that the harmonic function they have constructed is non-constant. Since the existence of such a non-constant harmonic function would contradict a Liouville type theorem proved by R. Schoen and S.-T. Yau [Comment. Math. Helv. 51, 333-341 (1976; Zbl 0361.53040)], the basis for the construction, i.e., the multiple ends, must not exist.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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