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Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds

Guido De Philippis, Alessandro Pigati

TL;DR

The paper proves a quantitative Michael–Simon inequality for anisotropic surface energies in dimension three, establishing that for $n=3$, $k=2$, and convex $F$ close to the isotropic area, a rectifiable $2$-varifold with finite mass and first variation satisfies $|V|(\\mathbb{R}^3)\\le C(F)\\mathcal{H}^2(\\{\\theta>0\\})^{1/2}|\\delta^F V|(\\mathbb{R}^3)$, where $\\theta$ is the lower density. The authors reduce the 3D problem to a new plane inequality, proving a sharp nonlinear bound for vector fields on the plane (a quantitative version of Alberti’s rank-one theorem) that leverages Smirnov-type decompositions and multilinear Kakeya-type ideas. They show that the Michael–Simon bound is equivalent to compactness of rectifiable varifolds under the Atomic Condition and extend the results to anisotropies including $\\ell^p$ norms, with a detailed analysis of the necessary convexity and positivity properties of the associated variation matrices. The work thus connects anisotropic geometric measure theory with plane Kakeya-type inequalities, providing stability results near the area and a framework for broader classes of anisotropic energies. The findings have significant implications for regularity and compactness in anisotropic variational problems.

Abstract

Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(Σ^k):=\int_ΣF(T_xΣ)\,d\mathcal{H}^k.$$ While a monotonicity formula is not available for critical points, when $k=2$ and $n=3$ we show that the Michael-Simon inequality holds if $F$ is convex and close to $1$ (in $C^1$), meaning that $\mathcal{F}$ is close to the usual area. Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem. As another byproduct, we also show Michael-Simon for another class of integrands which includes the $\ell^p$ norms for $p\in(1,\infty)$. For a general $F$ satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds.

Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds

TL;DR

The paper proves a quantitative Michael–Simon inequality for anisotropic surface energies in dimension three, establishing that for , , and convex close to the isotropic area, a rectifiable -varifold with finite mass and first variation satisfies , where is the lower density. The authors reduce the 3D problem to a new plane inequality, proving a sharp nonlinear bound for vector fields on the plane (a quantitative version of Alberti’s rank-one theorem) that leverages Smirnov-type decompositions and multilinear Kakeya-type ideas. They show that the Michael–Simon bound is equivalent to compactness of rectifiable varifolds under the Atomic Condition and extend the results to anisotropies including norms, with a detailed analysis of the necessary convexity and positivity properties of the associated variation matrices. The work thus connects anisotropic geometric measure theory with plane Kakeya-type inequalities, providing stability results near the area and a framework for broader classes of anisotropic energies. The findings have significant implications for regularity and compactness in anisotropic variational problems.

Abstract

Given an anisotropic integrand , we can generalize the classical isotropic area by looking at the functional While a monotonicity formula is not available for critical points, when and we show that the Michael-Simon inequality holds if is convex and close to (in ), meaning that is close to the usual area. Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem. As another byproduct, we also show Michael-Simon for another class of integrands which includes the norms for . For a general satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds.
Paper Structure (12 sections, 23 theorems, 145 equations)

This paper contains 12 sections, 23 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. New variants of multilinear Kakeya in dimension two
  4. General results in arbitrary dimension
  5. Preliminaries and proof of Propositions \ref{['equiv']}, \ref{['sumset']}, and \ref{['usc']}
  6. Nonlinear inequalities for vector fields on the plane
  7. Proof of the simpler \ref{['det.simple']}
  8. A more general bound
  9. Complementary inequalities
  10. Proof of \ref{['main.tw']}
  11. Proof of Michael--Simon for anisotropies close to the area
  12. Proof of \ref{['main.tris']}

Key Result

Theorem 1.3

If $n=3$, $k=2$, and $F$ is a convex integrand close enough to the isotropic area in the $C^1$ topology (i.e., $\|F|_{\mathbb{S}^{2}}-1\|_{C^1}$ is small enough), then the following holds. Given a rectifiable $2$-varifold $V$ in $\mathbb{R}^3$ with finite total mass and first variation, letting $\th In particular, ms holds for a possibly different constant $C(F)$.

Theorems & Definitions (40)

  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Proposition 1.9
  • Proposition 1.10
  • Corollary 1.11
  • ...and 30 more