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Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic First Variation Part II

Sławomir Kolasiński, Mario Santilli

Abstract

In a previous paper we proved quadratic flatness and $\mathscr{C}^{2}$-rectifiability for codimension one varifolds $ V $ on open subsets of $ \mathbf{R}^{n+1} $ with bounded anisotropic mean curvature, under the hypothesis that $ \operatorname{spt} \| V \| $ has locally finite $ \mathscr{H}^{n} $ measure. In this paper we explore some fundamental consequences of this result. In particular, we extend Brakke's perpendicularity theorem to the anisotropic setting and we deduce the locality of the anisotropic mean curvature vector.

Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic First Variation Part II

Abstract

In a previous paper we proved quadratic flatness and -rectifiability for codimension one varifolds on open subsets of with bounded anisotropic mean curvature, under the hypothesis that has locally finite measure. In this paper we explore some fundamental consequences of this result. In particular, we extend Brakke's perpendicularity theorem to the anisotropic setting and we deduce the locality of the anisotropic mean curvature vector.
Paper Structure (6 sections, 10 theorems, 163 equations)

This paper contains 6 sections, 10 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controlling tilt with height
  4. First variation of a push-forward of a varifold
  5. Perpendicularity and locality of the mean curvature
  6. Proof of the main theorem

Key Result

Theorem 1.1

Let $\phi$ be a uniformly convex $\mathscr{C}^{3}$-norm on $\mathbf{R}^{{n+1}}$ and let $F : \mathbf{G}(n+1,n) \rightarrow \mathbf{R}$ be the elliptic integrand naturally associated with $\phi$ (cf. mr:anisotropic_mean_curvature_vector). Suppose $\Omega \subseteq \mathbf{R}^{{n+1}}$ is open, $0 \leq Define $Q = \mathop{\mathrm{spt}}\nolimits \| V \| \cap \bigl\{ a : \boldsymbol{\Theta}^{n}(\| V \

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.4
  • proof
  • Definition 2.9
  • Remark 2.10
  • Remark 2.11
  • Definition 2.14
  • Definition 2.17
  • Remark 2.18
  • Definition 2.20
  • ...and 23 more