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$C^\infty$ rectifiability of stationary varifolds

Camillo Brena, Camillo De Lellis, Federico Franceschini

TL;DR

The paper proves that the support of any stationary $m$-dimensional integral rectifiable varifold in an open set $U\subset \mathbb{R}^{m+n}$ is $C^{\infty}$ rectifiable, i.e., covered up to $H^m$-null sets by countably many $C^{\infty}$ $m$-graphs. The authors innovate by replacing the classical best-approximating plane with a best-approximating minimal surface, establishing a decay lemma that drives $L^2$-distance contraction across scales. They develop a multivalued Lipschitz approximation on the normal bundle and a generalized tilt-excess inequality, together with elliptic regularization and Whitney-type arguments, to obtain $C^{\infty}$ rectifiability from quantitative excess decay. The results connect geometric measure theory with minimal-surface PDE techniques, yielding a robust route to higher-regularity structure of stationary varifolds and a framework potentially applicable to broader regularity problems in calibrated geometry and variational theory.

Abstract

In this paper we prove that, for every integers $m\geq 2$ and $n\geq 1$, the support of any stationary $m$-dimensional integer rectifiable varifold $V$ in an open set $U\subset \mathbb R^{m+n}$ is $C^\infty$ rectifiable, namely it can be covered, up to an $H^m$-null set, with countably many $C^\infty$ $m$-dimensional graphs.

$C^\infty$ rectifiability of stationary varifolds

TL;DR

The paper proves that the support of any stationary -dimensional integral rectifiable varifold in an open set is rectifiable, i.e., covered up to -null sets by countably many -graphs. The authors innovate by replacing the classical best-approximating plane with a best-approximating minimal surface, establishing a decay lemma that drives -distance contraction across scales. They develop a multivalued Lipschitz approximation on the normal bundle and a generalized tilt-excess inequality, together with elliptic regularization and Whitney-type arguments, to obtain rectifiability from quantitative excess decay. The results connect geometric measure theory with minimal-surface PDE techniques, yielding a robust route to higher-regularity structure of stationary varifolds and a framework potentially applicable to broader regularity problems in calibrated geometry and variational theory.

Abstract

In this paper we prove that, for every integers and , the support of any stationary -dimensional integer rectifiable varifold in an open set is rectifiable, namely it can be covered, up to an -null set, with countably many -dimensional graphs.

Paper Structure

This paper contains 27 sections, 21 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Strategy of the proof
  4. Decay lemma
  5. Elliptic estimates for the distance from a minimal surface
  6. Lipschitz approximation on the normal bundle
  7. Preliminaries
  8. The geometric set-up
  9. Sections of the normal bundle of $M$
  10. The Jacobi operator
  11. The minimal surface system and its linearization
  12. Some results on stationary varifolds
  13. Proof of the tilt excess inequality and of the height bound
  14. Proof of Propositions \ref{['prop:caccioppoli']}
  15. Proof of Proposition \ref{['prop:altezza']}
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $V$ be an $m$-dimensional stationary varifold in an open set $U\subset \mathbb R^{m+n}$. Then $V$ is $C^{\infty}$-rectifiable. Namely, there exist countably many $C^{\infty}$ maps $f_k:\mathbb{R}^m\supset B_1^m\rightarrow U$ such that

Theorems & Definitions (35)

  • Conjecture 1.1
  • Theorem 1.2: Smooth rectifiability
  • Theorem 1.3
  • Definition 2.1: $\delta$-flatness
  • Lemma 2.2: Decay lemma
  • Lemma 2.3
  • Remark 2.4
  • Proposition 2.5: Tilt-excess inequality
  • Proposition 2.6: Height bound
  • Remark 2.7
  • ...and 25 more