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

Xuwen Zhang

Published 2021-10-19, updated 2022-05-05Version 3

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 distance function from the boundary of any Borel set in sphere, are $C^{1,1}$-rectifiable. As a by-product, we establish a Heintze-Karcher inequality.