Absolutely area minimizing singular cones of arbitrary codimension

Bindschadler, David

doi:10.1090/S0002-9947-1978-0487726-1

Absolutely area minimizing singular cones of arbitrary codimension
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Trans. Amer. Math. Soc. 243 (1978), 223-233 Request permission

Abstract:

The examples of area minimizing singular cones of codimension one discovered by Bombieri, DeGiorgi and Guisti are generalized to arbitrary codimension, thus filling a dimensional gap. Previously the only nontrivial examples of singular area minimizing integral currents of codimension other than one were obtained from holomorphic varieties and hence of even codimension. Specifically, let S be the N-fold Cartesian product of p-dimensional spheres and C be the cone over S. We prove that for p sufficiently large, C is absolutely area minimizing. It follows from the technique used that C restricted to the ball of radius ${N^{1/2}}$ is the unique solution to the oriented Plateau problem with boundary S.

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