An optimal extension of Perelman's comparison theorem for quadrangles and its applications
Abstract
In this paper we discuss an extension of Perelman's comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with non-negative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become totally convex relative to a bigger sub-domain. An optimal extension of 2nd variational formula for geodesics by Petrunin will be derived for the case of non-negative curvature. In addition, we also introduced the generalized second fundament forms for subsets in Alexandrov spaces. Using this new notion, we will propose an approach to study two open problems in Alexandrov geometry.
Keywords
Cite
@article{arxiv.0712.3221,
title = {An optimal extension of Perelman's comparison theorem for quadrangles and its applications},
author = {Jianguo Cao and Bo Dai and Jiaqiang Mei},
journal= {arXiv preprint arXiv:0712.3221},
year = {2009}
}
Comments
We corrected some inaccurate statements and definitions about development maps related to Corollary 2.4, based on Professor Stephanie Alexander's suggestions