A Slope-Bounding Procedure for Mass-Minimizing Currents

Abstract

In the higher codimensional, non-parametric context the area integrand is not a convex function, but if restricted to small enough slopes the integrand is convex. That non-convexity prevents the extension to higher codimensions of certain methods applicable to the problem of explicit determination of area-minimizing hypersurfaces. We give a simple proof that area-minimizing surfaces with small excess are graphs with small slope. We also introduce a strategy for a posteriori estimation of the excess—a strategy applicable in the presence of a rich enough family of approximating surfaces. It is hoped that it will be possible to combine the a posteriori excess estimation strategy and the slope-bounding result so as to extend to higher codimensions the explicit determination methods that are now limited to hypersurfaces.