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Remarks and Conjectures on Stationary Varifolds

Camillo Brena, Stefano Decio, Camillo De Lellis

TL;DR

The paper develops a streamlined framework for the interior regularity of stationary integer-rectifiable varifolds by proving a sequence of height bounds, Lipschitz approximations, and excess-decay estimates. The approach leverages a multivalued (Q-valued) Lipschitz graph representation and harmonic-approximation techniques to obtain sharp control on the varifold near a reference plane, leading to almost quadratic excess decay and C^{1,1−δ} rectifiability. It also analyzes optimality through catenoid examples, connects these results to classic comparisons in the literature, and outlines a program toward resolving conjectures on singular sets and unique continuation in stationary varifolds. Overall, the work provides a transparent, self-contained path to refined regularity results and sharp quantitative estimates relevant for the broader study of geometric measure theory and minimal-surface-like variational problems.

Abstract

In this paper, we revisit some known results about stationary varifolds using simpler arguments. In particular, we obtain the height bound and the Lipschitz approximation along with its estimates, and as a consequence, the excess decay

Remarks and Conjectures on Stationary Varifolds

TL;DR

The paper develops a streamlined framework for the interior regularity of stationary integer-rectifiable varifolds by proving a sequence of height bounds, Lipschitz approximations, and excess-decay estimates. The approach leverages a multivalued (Q-valued) Lipschitz graph representation and harmonic-approximation techniques to obtain sharp control on the varifold near a reference plane, leading to almost quadratic excess decay and C^{1,1−δ} rectifiability. It also analyzes optimality through catenoid examples, connects these results to classic comparisons in the literature, and outlines a program toward resolving conjectures on singular sets and unique continuation in stationary varifolds. Overall, the work provides a transparent, self-contained path to refined regularity results and sharp quantitative estimates relevant for the broader study of geometric measure theory and minimal-surface-like variational problems.

Abstract

In this paper, we revisit some known results about stationary varifolds using simpler arguments. In particular, we obtain the height bound and the Lipschitz approximation along with its estimates, and as a consequence, the excess decay

Paper Structure

This paper contains 13 sections, 14 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Conjectures
  3. Rectifiability
  4. Main statements
  5. Catenoids
  6. Comparison with the existing literature
  7. Preliminaries
  8. Proof of the main results
  9. First height bound
  10. First Lipschitz approximation
  11. Improved height bound
  12. Improved Lipschitz approximation
  13. Excess decay

Key Result

Theorem 1.8

Assume $V$ and $x_0$ satisfy the assumptions of Conjecture c:flat=regular. Then there is a smooth classical (not necessarily minimal) $m$-dimensional graph $\mathcal{M}$ in some neighborhood of $x_0$ with the property that

Theorems & Definitions (34)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Definition 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Theorem 1.9: Height bound
  • Definition 1.10
  • ...and 24 more