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

TL;DR

This paper proves that level-sets of distance functions in spheres are $C^{1,1}$-rectifiable and establishes a Heintze-Karcher inequality on the sphere using nonlinear analysis on embedded graphs.

Contribution

It introduces a novel approach by embedding spheres into Euclidean space and analyzing codimension-2 graphs to prove rectifiability and derive geometric inequalities.

Findings

Level-sets of distance functions are $C^{1,1}$-rectifiable.

Established a Heintze-Karcher inequality on the sphere.

Applied nonlinear analysis on embedded graphs to geometric problems.

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.