Solutions to the minimal surface system with large singular sets

TL;DR

This paper investigates the limits of area-minimizing sequences for minimal surfaces in higher codimension, revealing large singular sets and discrepancies between parametric and non-parametric problems, even in low dimensions.

Contribution

It constructs explicit examples showing large singular sets in limits of Lipschitz minimal surface sequences in the lowest possible dimension and codimension.

Findings

Limits can have large interior vertical and non-minimal parts

Demonstrates discrepancy between parametric and non-parametric minimization

Constructs examples in dimension 3 and codimension 2

Abstract

Lawson and Osserman proved that the Dirichlet problem for the minimal surface system is not always solvable in the class of Lipschitz maps. However, it is known that minimizing sequences (for area) of Lipschitz graphs converge to objects called Cartesian currents. Essentially nothing is known about these limits. We show that such limits can have surprisingly large interior vertical and non-minimal portions. This demonstrates a striking discrepancy between the parametric and non-parametric area minimization problems in higher codimension. Moreover, our construction has the smallest possible dimension ($n = 3$) and codimension $(m = 2)$.