On the singularities and a Bernstein property of immersed stable minimal hypersurfaces
This work is a study of the singularities of weak limits of immersed stable minimal hypersurfaces of arbitrary dimension, leading to a Bernstein type result. It is a follow up of a separate work by the author to appear in the Journal of Differential Geometry, and makes use of the Allard regularity theorem and results of Schoen, Simon and Yau. The arguments involved are based mostly on compactness properties of stationary varifolds and the dimension reducing principle. Letting \(I\) denote the family of oriented smooth immersed stable minimal hypersurfaces of finite volume in the unit ball of \((n+1)\)-dimensional Euclidean space, \(\overline{I}\) denotes the closure of \(I\) in the sense of varifolds. If \(V\) is a varifold in \(\overline{I}\), the singularities of \(V\) are studied when the density of \(V\) is not much larger than 2, for example, it is shown that there exists a constant \(e\) between \(0\) and \(1\) and depending only on n so that, if at generic points in the singular set of \(V\) the density is at most \(2+e\), then all tangent cones to \(V\) are pairs of hyperplanes (where the density is then exactly 2). After this, a Bernstein type result is given: Let \(w_n\) denote the volume of the unit ball in \(n\)-dimensional Euclidean space. Let \(n\) be between 2 and 6. Let \(M\) be a complete non-compact, stable minimal hypersurface immersed in \((n+1)\)-dimensional Euclidean space. Let \(H_n(S)\) denote the \(n\)-dimensional Hausdorff dimension of the set \(S = M\) intersect the \((n+1)\)-dimensional radius \(R\) ball in \((n+1)\)-dimensional Euclidean space centered at some point \(x\). Then there exists an absolute positive constant \(b\) so that, if \(H_n(S)/(w_n\) times the \(n\)th power of \(R\)) is at most \(2+b\) for some \(x\) in \((n+1)\)-dimensional Euclidean space for all positive \(R\), then \(M\) must be either a hyperplane or a pair of parallel or transverse hyperplanes. A similar slightly more general result is given in the case when \(M\) is replaced by a varifold \(V\).
scientific article; zbMATH DE number 2135753
scientific article