正标量曲率几何中的刚性结果
微分几何
2025-12-30 v5 广义相对论与量子宇宙学
数学物理
math.MP
摘要
我证明了球面的标量曲率刚性定理。特别地，我证明了在 维单位球面中，半径严格小于 的测地球在光滑变形下可以是刚性的，这些变形在增加标量曲率的同时保持内蕴几何和边界的平均曲率，而这种刚性结果对半球不成立。该断言的证明需要实 Killing 联络的概念及其 Dirac 算子相关边值问题的解。该结果是对现已证伪的 Min-Oo 猜想的最精确细化。
引用
@article{arxiv.2410.23406,
title = {On a Rigidity Result in Positive Scalar Curvature Geometry},
author = {Puskar Mondal},
journal= {arXiv preprint arXiv:2410.23406},
year = {2025}
}
备注
The result is far from optimal; the radius of the geodesic ball that can be rigid under large deformation is not clear at all, making the result almost useless despite some new techniques such as using the biorthogonal spin system. Therefore, it is best to retract this