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Uniqueness in the Plateau problem for calibrated currents

Bryan Dimler, Chen-Kuan Lee

TL;DR

The paper proves that compactly supported, smoothly calibrated integral currents with connected $C^{3,\alpha}$ boundary are uniquely determined by their boundary data in the oriented Plateau problem, by combining boundary regularity theory for area-minimizing currents with Morgan-type unique continuation from Cauchy data. The approach hinges on calibrations (via a comass-one closed form) to enforce mass-minimization, a boundary regularity framework that yields a dense set of one-sided boundary points, and a boundary-tangent-space alignment argument (First Cousin Principle and Intersecting Planes) to propagate equality from the boundary inward. The results recover known uniqueness cases, extend to higher codimensions and flat chains, and establish sufficient regularity conditions on calibrations in codimension one, with sharpness analyses for the hypotheses. The methods and conclusions contribute a general, dimension/codimension-free criterion for uniqueness in the oriented Plateau problem, with extensions to ambient manifolds under topological constraints such as $H_k(M;\mathbb{R})=0$.

Abstract

We show that every compactly supported smoothly calibrated integral current with connected $C^{3,α}$ boundary is the unique solution to the oriented Plateau problem for its boundary data. The same holds true for compactly supported $``$continuously calibrated$"$ integral flat chains. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and a unique continuation argument in the spirit of Frank Morgan. In codimension one, the argument yields a sufficient condition for uniqueness in the oriented Plateau problem expressed in terms of the regularity of the calibrating form.

Uniqueness in the Plateau problem for calibrated currents

TL;DR

The paper proves that compactly supported, smoothly calibrated integral currents with connected boundary are uniquely determined by their boundary data in the oriented Plateau problem, by combining boundary regularity theory for area-minimizing currents with Morgan-type unique continuation from Cauchy data. The approach hinges on calibrations (via a comass-one closed form) to enforce mass-minimization, a boundary regularity framework that yields a dense set of one-sided boundary points, and a boundary-tangent-space alignment argument (First Cousin Principle and Intersecting Planes) to propagate equality from the boundary inward. The results recover known uniqueness cases, extend to higher codimensions and flat chains, and establish sufficient regularity conditions on calibrations in codimension one, with sharpness analyses for the hypotheses. The methods and conclusions contribute a general, dimension/codimension-free criterion for uniqueness in the oriented Plateau problem, with extensions to ambient manifolds under topological constraints such as .

Abstract

We show that every compactly supported smoothly calibrated integral current with connected boundary is the unique solution to the oriented Plateau problem for its boundary data. The same holds true for compactly supported continuously calibrated integral flat chains. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and a unique continuation argument in the spirit of Frank Morgan. In codimension one, the argument yields a sufficient condition for uniqueness in the oriented Plateau problem expressed in terms of the regularity of the calibrating form.

Paper Structure

This paper contains 19 sections, 16 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal surface system
  4. Integral currents
  5. Area-minimizing integral currents
  6. Rectifiable varifolds
  7. Densities
  8. Smoothly calibrated integral currents
  9. Unique continuation from Cauchy data
  10. Lipschitz stationary solutions
  11. Area-minimizing currents
  12. Uniqueness of compactly supported smoothly calibrated currents
  13. Proof of TEXT
  14. Sharpness of hypotheses
  15. Connectedness of the boundary
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $U = \mathbb{R}^n$ and suppose that $T \in \mathcal{I}_{k,c}(\mathbb{R}^n)$ and $\Gamma$ are as in A1. Let $\varphi \in \mathcal{E}^k(\mathbb{R}^n)$ (i.e. smooth $k$-form in $\mathbb{R}^n$) be a calibration. Assume that $T$ is calibrated by $\varphi$ in $\mathbb{R}^n$. If $T^\prime \in \mathcal{

Theorems & Definitions (46)

  • Theorem 1.2: Uniqueness in the Oriented Plateau Problem
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Integral Current
  • Remark 2.2
  • Example 2.3: Lipschitz Submanifolds
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7: cf. HL82
  • ...and 36 more