Calibrating Forms for Minimal Graphs in Arbitrary Codimension

TL;DR

This paper introduces a new family of differential forms associated with minimal graphs in any codimension, providing a criterion for their area-minimizing property and confirming a conjecture of Lawson and Osserman.

Contribution

It constructs explicit non-parallel calibrations for minimal graphs in arbitrary codimension, extending classical theory and offering a new criterion for area-minimization.

Findings

Constructed explicit closed forms for minimal graphs in arbitrary codimension.

Characterized comass bounds via singular values and two-dilations.

Confirmed Lawson-Osserman conjecture under two-dilation conditions.

Abstract

We introduce a new family of closed differential forms naturally associated with minimal graphical submanifolds in Euclidean space, defined in arbitrary codimension. For each minimal graph, we construct an explicit closed form whose restriction coincides with the induced volume form. These forms admit a geometric interpretation as pullbacks, via the Gauss map, of tautological differential forms on the Grassmannian. In contrast to most known calibrations, they are generally not parallel and do not arise from special holonomy or symmetry considerations. The calibration problem is thus reduced to estimating the pointwise comass of the constructed forms. We show that the comass bound can be characterized in terms of explicit inequalities involving the singular values of the defining map of the graph, formulated via its two-dilations and we identify precise conditions ensuring that the comass is at most one. As a consequence, any minimal graph satisfying these conditions is calibrated and hence area-minimizing. This yields a broad class of new calibrated minimal graphs, extending the classical codimension-one theory, and provides an effective criterion for determining precisely where a given minimal graph is area-minimizing. As an application of our construction, we confirm a conjecture of Lawson and Osserman under two-dilation conditions, in arbitrary codimesnion.