Proof of Bishop's volume comparison theorem using singular soap bubbles
Differential Geometry
2019-04-01 v1
Abstract
Bishop's volume comparison theorem states that a compact -manifold with Ricci curvature larger than the standard -sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome.
Cite
@article{arxiv.1903.12317,
title = {Proof of Bishop's volume comparison theorem using singular soap bubbles},
author = {Hubert Bray and Feng Gui and Zhenhua Liu and Yiyue Zhang},
journal= {arXiv preprint arXiv:1903.12317},
year = {2019}
}
Comments
13 pages, 6 figures