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

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.

23 pages, 2 figures

References in corpus (3)

• Alexandrov's theorem revisited
• Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets
• Fine properties of the curvature of arbitrary closed sets