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On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities

Zhenhua Liu

TL;DR

The work settles Almgren's conjecture by constructing $d$-dimensional area-minimizing currents on smooth compact manifolds whose singular sets can have prescribed fractal (Hausdorff) dimensions $0\le\alpha\le d-2$, with the singular strata aligned to prescribed Almgren strata. The construction hinges on calibrations designed on product manifolds, using a two-piece current $X\times N_0+Y\times N_f$ and a function $f$ vanishing to infinite order on a target fractal set, engineered via a tailored metric $g_N$ and retractions. The results extend to area-minimizing currents modulo $v$ and to stable stationary varifolds, establishing the sharpness of the rectifiability bounds and enabling explicit prescriptions of the strata. The methods provide a robust framework for realizing fractal singularities in high-codimension area-minimizing objects, with implications for the structure of singular sets in geometric measure theory and calibrations. The work also discusses limitations in analytic settings, compares with related results (e.g., Simon’s), and outlines directions for further generalizations and interactions among strata.

Abstract

We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle a conjecture by Almgren and our answer is sharp dimensionwise. Furthermore, we can prescribe arbitrarily the strata in the Almgren stratification of the singular sets of our area-minimizing submanifolds, and our results hold in the category of integral currents, mod v currents and stable stationary varifolds.

On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities

TL;DR

The work settles Almgren's conjecture by constructing -dimensional area-minimizing currents on smooth compact manifolds whose singular sets can have prescribed fractal (Hausdorff) dimensions , with the singular strata aligned to prescribed Almgren strata. The construction hinges on calibrations designed on product manifolds, using a two-piece current and a function vanishing to infinite order on a target fractal set, engineered via a tailored metric and retractions. The results extend to area-minimizing currents modulo and to stable stationary varifolds, establishing the sharpness of the rectifiability bounds and enabling explicit prescriptions of the strata. The methods provide a robust framework for realizing fractal singularities in high-codimension area-minimizing objects, with implications for the structure of singular sets in geometric measure theory and calibrations. The work also discusses limitations in analytic settings, compares with related results (e.g., Simon’s), and outlines directions for further generalizations and interactions among strata.

Abstract

We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle a conjecture by Almgren and our answer is sharp dimensionwise. Furthermore, we can prescribe arbitrarily the strata in the Almgren stratification of the singular sets of our area-minimizing submanifolds, and our results hold in the category of integral currents, mod v currents and stable stationary varifolds.

Paper Structure

This paper contains 39 sections, 15 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Plan of proof
  3. Overview of the paper
  4. Basic definitions
  5. Euclidean spaces and flat tori
  6. Manifolds
  7. Riemannian geometry
  8. Planes, vectors and forms
  9. Several facts about comass
  10. A torus differential form
  11. Product of simple forms
  12. The normal bundle calibration
  13. Definition of area-minimizing
  14. Definition of singular sets
  15. Area-minimizing integral currents in Riemannian products and disjoint unions
  16. ...and 24 more sections

Key Result

Theorem 1

For any integer $d\ge 2$ and any nonnegative real number $0\le\alpha\le d-2,$ there exists a smooth compact $(d+3)$-dimensional Riemannian manifold $M^{d+3}$, and a $d$-dimensional area-minimizing integral current $\Sigma$ on $M$ such that

Figures (1)

  • Figure 1: Pushing $\left\llbracket y_1y_2x_3\right\rrbracket$ into $y_3$ direction parametrically by $f$

Theorems & Definitions (35)

  • Conjecture 1
  • Theorem 1
  • Definition 1.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.3
  • Lemma 1.4
  • Lemma 2.2
  • proof
  • ...and 25 more