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$C^{1,1}$-rectifiability and Heintze-Karcher inequality on $\mathbf{S}^{n+1}$

Xuwen Zhang

Abstract

In this paper, by isometrically embedding $(\mathbf{S}^{n+1},g_{\mathbf{S}^{n+1}})$ into $\mathbf{R}^{n+2}$, and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open set in sphere, are $C^{1,1}$-rectifiable. As a by-product, we establish a Heintze-Karcher inequality on sphere.

$C^{1,1}$-rectifiability and Heintze-Karcher inequality on $\mathbf{S}^{n+1}$

Abstract

In this paper, by isometrically embedding into , and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open set in sphere, are -rectifiable. As a by-product, we establish a Heintze-Karcher inequality on sphere.

Paper Structure

This paper contains 12 sections, 15 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Acknowledgements
  5. Rectifiable sets
  6. Rectifiable set
  7. Area formula and Coarea formula
  8. Geometry of S,g
  9. Fine Properties of Gamma-s-t
  10. C11-rectifiability of Gamma-s-t
  11. The Heintze-Karcher inequality on sphere
  12. Codimension-2 graphs

Key Result

Theorem \oldthetheorem

Among sets of finite perimeter and finite volume, finite unions of balls with equal radii are the unique critical points of the Euclidean isoperimetric problem.

Figures (2)

  • Figure 1: Relation of $x,y,z$ and $N(y)$
  • Figure 2: Relation of $y$, $g_r(y)$ and the corresponding normals.

Theorems & Definitions (34)

  • Theorem \oldthetheorem: DM19
  • Proposition \oldthetheorem: DM19
  • Definition \oldthetheorem: $\Gamma_s^t$ and $\Gamma_s^+$
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Heintze-Karcher inequality on Sphere
  • Definition \oldthetheorem: $n$-rectifiable set, DM19
  • Proposition \oldthetheorem: Area formula for $k$-rectifiable sets, Mag12,Sim83
  • Proposition \oldthetheorem: Coarea formula for $k$-rectifiable set, Sim83
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • ...and 24 more