关于渐近平坦流形和渐近双曲流形上一类新质量的相关问题的研究

Positive mass conjecture and Penrose inequality play an essential role in general relativity and also are important objects in mathematics. This project is concerned with the properties of Gauss-Bonnet-Chern mass for asymptotically flat manifolds and asymptotically hyperbolic manifolds respectively, which is a new kind of mass introduced recently by the applicant and her cooperators. As the generalization of ADM mass, the study of positive mass theorem and Penrose inequalities for GBC mass is basic and also important in Mathematics as well as Physics. Due to the complexity of GBC mass, this project will mainly concentrate on the graphical manifolds first. We will mainly consider the following three problems:.(1) Regarding the positivity of GBC mass and its corresponding Penrose inequalities on graphical asymptotically flat manifolds of high codimension..(2) Weighted Alexandrov-Fenchel inequalities in the hyperbolic space. They are a class of very important geometric inequalities and have a close relationship with the Penrose inequalities related to GBC mass for asymptotically hyperbolic graphs..(3) Study the rigidity problem of positive mass theorem and Penrose inequalities of GBC mass for asymptotically flat graphs and asymptotically hyperbolic graphs .. Through the study of this project, further understanding of positive mass conjecture and Penrose inequality for general asymptotically flat manifolds and asymptotically hyperbolic manifolds will be deeply expected.

正质量猜想和Penrose不等式是广义相对论中一个核心问题，也是数学上的重要研究对象。本项目将针对申请人和合作者最近在渐近平坦流形和渐近双曲流形上提出的一类新质量：Gauss-Bonnet-Chern(GBC)质量，展开其正质量猜想和Penrose不等式的研究。由于一般情形的困难性，本项目将集中于图流形上GBC质量的性质，拟研究如下问题：.(1)高余维的渐近平坦图流形上关于GBC质量的正性和Penrose不等式。.(2) 双曲空间上的加权的Alexandrov-Fenchel不等式。这是一类非常重要的几何不等式，与渐近双曲图流形上的Penrose不等式有着非常密切的关系。.(3) 渐近平坦图流形和渐近双曲图流形上关于GBC质量的正质量定理刚性和Penrose不等式的刚性问题。 . 本项目的研究将加深对正质量猜想和Penrose不等式的理解，并期望有助于一般流形上相关问题的研究。

本项目按计划顺利完成，已经完成SCI收录学术论文4篇和1篇国内核心刊物论文，发表于Math. Z，Pacific J. Math., Cal. Var. , Proc. Amer. Math. Soc. 等期刊。我们主要研究了渐近平坦流形和渐近双曲流形中的Gauss-Bonnet-Chern(GBC)质量性质及相关的几何不等式问题以及超曲面的刚性问题。我们在渐近平坦流形上利用推广的k-Einsten张量得到内蕴定义的高阶质量，并通过陈形式引入了质量的另一种定义，并证明了这两种不同定义下的质量与GBC质量的等价性；利用逆平均曲率流，证明了双曲空间中关于4阶平均曲率积分的Alexandrov-Fenchel型不等式，并且将以前的双曲空间上的Alexandrov-Fenchel型不等式推广到外围流形为Kottler空间情形。该结果的一个重要应用是可以建立Kottler空间图流形上的最优的Penrose不等式；关于超曲面的刚性问题的研究，我们首先给出了标准双曲空间测地球面的一个新的刻画，并进一步研究当外围空间是满足一定条件的warped 积流形空间，并且考虑关于内蕴的的Gauss-Bonnet曲率及其全积分第一变分出来的项的商条件的刚性问题。