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Critical Allard regularity: pointwise tilt-excess estimates

Sean McCurdy

Abstract

The main results of this paper provide VMO-type estimates for the quadratic tilt-excess on varifolds with critical generalized mean curvature. These estimates apply to varifolds with "almost-integral" density which are close to a multiplicity one $m$-disc in a ball in the usual senses. The class of almost-integral varifolds allows for varifolds with non-perpendicular mean curvature. Moreover, the estimates hold \emph{uniformly for every point} in a relatively open set in $\text{spt}||V||$ and naturally imply a Reifenberg-type parametrization. The proof relies upon generalizing the $Q$-valued Lipschitz approximation and Sobolev-Poincaré estimates of arXiv:0808.3660 to almost-integral rectifiable varifolds.

Critical Allard regularity: pointwise tilt-excess estimates

Abstract

The main results of this paper provide VMO-type estimates for the quadratic tilt-excess on varifolds with critical generalized mean curvature. These estimates apply to varifolds with "almost-integral" density which are close to a multiplicity one -disc in a ball in the usual senses. The class of almost-integral varifolds allows for varifolds with non-perpendicular mean curvature. Moreover, the estimates hold \emph{uniformly for every point} in a relatively open set in and naturally imply a Reifenberg-type parametrization. The proof relies upon generalizing the -valued Lipschitz approximation and Sobolev-Poincaré estimates of arXiv:0808.3660 to almost-integral rectifiable varifolds.
Paper Structure (26 sections, 30 theorems, 189 equations)

This paper contains 26 sections, 30 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. The situation to be studied
  3. The results of the present paper.
  4. Discussion of the proofs and outline of the paper.
  5. Notation
  6. Preliminaries
  7. Compactness
  8. Proof of Theorem \ref{['t:rect varifolds compactness']}
  9. Basic results for rectifiable varifolds.
  10. Harmonic and almost-weakly harmonic functions
  11. Lipschitz Approximation
  12. Results preliminary to the Lipschitz approximation
  13. Density estimates
  14. The Limit Case
  15. Lemmata preliminary to the Lipschitz approximation
  16. ...and 11 more sections

Key Result

Theorem 1.3

(Zhou_WillmoreEnergy22) Let $m=p=2$ and $n,V$ be as in s: the set up. Let $0<\tilde{\gamma}$, $0<\alpha<1$, and suppose that $V$ satisfies the hypotheses of the $(2,2,\tilde{\gamma})$-Allard regularity regime in $B_1^{n}(0)$. If $\boldsymbol{h}(V, x)$ is perpendicular to $\emph{ap Tan}_x \emph{spt}\ where $\rho = \rho(n) \in (0,1/2)$.

Theorems & Definitions (74)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.9
  • Definition 1.10
  • Theorem 1.11
  • ...and 64 more