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A monotonicity formula for anisotropic minimal hypersurfaces

Doanh Pham

TL;DR

The paper addresses monotonicity for anisotropic minimal hypersurfaces by embedding the problem in an equiaffine, anisotropic framework. It derives a general monotonicity identity in affine geometry using a tangential projection and a carefully chosen tangent field, then specializes to the anisotropic setting with the anisotropic normal $\nu_F$ and the dual norm $F^{\circ}$ to obtain a monotone energy $r \mapsto (1/r^n) \int_{M\cap (r\Omega)} F(\nu)$ under Condition S. This yields a sharp area lower bound for $F$-minimal hypersurfaces in the Wulff shape, with rigidity described by hyperplanes, and ties to existing anisotropic PDE results via Condition S. The appendix extends these ideas to affine $k$-curvatures, providing Minkowski-type formulas. Overall, the work extends classical isotropic monotonicity to anisotropic settings, offering new tools for regularity and variational problems in anisotropic geometries.

Abstract

Under a sign assumption on the Minkowski norm, we prove a monotonicity formula for anisotropic minimal hypersurfaces in Euclidean space.

A monotonicity formula for anisotropic minimal hypersurfaces

TL;DR

The paper addresses monotonicity for anisotropic minimal hypersurfaces by embedding the problem in an equiaffine, anisotropic framework. It derives a general monotonicity identity in affine geometry using a tangential projection and a carefully chosen tangent field, then specializes to the anisotropic setting with the anisotropic normal and the dual norm to obtain a monotone energy under Condition S. This yields a sharp area lower bound for -minimal hypersurfaces in the Wulff shape, with rigidity described by hyperplanes, and ties to existing anisotropic PDE results via Condition S. The appendix extends these ideas to affine -curvatures, providing Minkowski-type formulas. Overall, the work extends classical isotropic monotonicity to anisotropic settings, offering new tools for regularity and variational problems in anisotropic geometries.

Abstract

Under a sign assumption on the Minkowski norm, we prove a monotonicity formula for anisotropic minimal hypersurfaces in Euclidean space.
Paper Structure (9 sections, 10 theorems, 52 equations)

This paper contains 9 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minkowski norms and Wulff shapes
  4. Affine geometry
  5. Anisotropic geometry
  6. Monotonicity formula
  7. Minkowski-type formula for equiaffine hypersurfaces
  8. Elementary symmetric functions and Newton tensors
  9. Minkowski-type formula

Key Result

Theorem 1.1

Let $M$ be an oriented smooth hypersurface of $\mathbb{R}^{n+1}$ with a chosen unit normal vector field $\nu$. Suppose that $M$ is $F$-minimal and satisfies $\partial M \subset \bar{r}\mathcal{W}$ for some $\bar{r} > 0$. Then, for every $0 < s < r < \bar{r}$, we have In particular, if $F$ satisfies Condition S eq: sign condition for F, then the normalized anisotropic energy is monotone increasin

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 13 more