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Calibrating Forms for Minimal Graphs in Arbitrary Codimension

Chung-Jun Tsai, Mu-Tao Wang

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.

Calibrating Forms for Minimal Graphs in Arbitrary Codimension

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.

Paper Structure

This paper contains 8 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Differential Forms Associated with a Graph
  3. The definition of $\Theta(F)$
  4. The Closedness Condition
  5. A Coordinate Expression of $\Theta(F)$
  6. The closedness of $\Theta(F)$ and the minimal graph system
  7. Estimating the Comass
  8. Lipschitz Solution to the Minimal Graph System

Key Result

Theorem 1.4

Let $F:\Omega\subset \mathbb{R}^n\rightarrow \mathbb{R}^2$ be a smooth map whose graph is a minimal submanifold of $\mathbb{R}^{n+2}$. Suppose that $F$ is area-non-increasing, i.e. its $2$-dilation is no greater than one: where $\{\lambda_i\}_{i=1,2}$ are the singular values of ${\mathrm d} F$. Then the graph of $F$ is calibrated by $\Theta(F)$, defined in Definition defn:pre-calibration$\blacktr

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 17 more