A note on the singular set of area-minimizing hypersurfaces

Abstract

We give an a priori bound on the (n − 7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.Area-minimizing surfaces in general will not be smooth, and a basic question in minimal surface theory is to understand the size and nature of the singular region. The cumulative works of many (Federer, De Giorgi, Allard, Simons, to name only a few) prove that for absolutelyarea-minimizing n-dimensional hypersurfaces in R n+1 ("codimensionone area-minimizing integral currents"), the interior singular set is at most (n − 7)-dimensional. This dimension bound is sharp, and is directly tied to the existence of low-dimensional, non-flat minimizing cones.[HS79] proved that for such codimension-one area-minimizers, if the boundary is known to be C 1,α and multiplicity-one, then in fact no singularities lie within a neighborhood of the boundary. Combined with interior regularity, this theorem gives a very nice structure of these minimizing hypersurfaces.Recently [NV17], [NV15] quantified the interior partial regularity, by demonstrating effective local (interior) bounds on the H n−7 measure of the singular set. Their methods also prove (n − 7)-rectifiability of the singular set, which was originally established through an entirely different approach by [Sim95].In this short note, we obtain obtain a global, effective a priori estimate on the singular set of an area-minimzing hypersurface in terms of the boundary geometry. Our results are loosely analogous to the a priori bounds of [AL88] (see also the recent works [MMS18b], [MMS18a]).We work in R n+1 , for n ≥ 7. Let us write I n (U) for the space of integral n-currents acting on forms supported in the open set U. Given an n-dimensional, oriented manifold E, write [E] for the current induced by integration. Let η λ (x) = λx, and τ y (x) = x + y.If T ∈ I n (U), we say T is area-minimizing if ||T ||(W ) ≤ ||T +S||(W ) for every open W ⊂⊂ U, and every S ∈ I n (U) satisfying ∂S = 0,

A Bernstein-type theorem for minimal graphs over convex domains

“…Half of Enneper's surface, and the half-helicoid, are both area-minimizing hypersurfaces in R 3 which bound a line ( [22]; cf. [8]). Neither is contained in a half-space.…”