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Sharp lower bounds on density for area-minimizing cones. (English) Zbl 1341.53012

In this paper, the authors consider the question for the infimum for density among all area-minimizing hypercones \(C \subset \mathbb{R}^{n}\) with an isolated singularity at the origin. This question is equivalent to the question of the infimum for density among all pairs \((M,x)\), where \(M\) is an area-minimizing hypersurface in a Riemannian manifold and \(x\) is an interior singular point of \(M\). Specifically, they provide a sharp answer to this question under a topological assumption, which amounts to a positive answer to a conjecture of Bruce Solomon in the case when one considers the (particularly prevalent examples of) hypercones that satisfy the assumption:
Theorem. Suppose that \(C \subset \mathbb{R}^{n}\) is an area-minimizing hypercone with an isolated singularity at the origin, and also that at least one of the two components of \(\mathbb{R}^{n} \setminus C\) is non-contractible. Then the density of \(C\) at the origin is greater than \(\sqrt{2}\). An equivalent formulation for minimal submanifolds of spheres is given. The proof relies on several facts about mean curvature flow.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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