分数阶数量曲率的相关问题研究

Fractional scalar curvature is a nonlocal geometric invariant, which is derived from the study of the fractional Yamabe problems. In this project, we will focus on the study of the compactness of the solutions to the fractional Yamabe problems which is related to the fractional scalar curvature, and some rigidity of the Riemann manifolds with nonnegative constant fractional scalar curvature. We begin with the ideas of researches on the classical positive mass theorem, then try to formulate and prove the fractional type positive mass theorem. By considering the degenerate elliptic PDE with Dirichlet-to-Neumann boundary condition which is equivalent to the fractional Yamabe problem, and also fractional type positive mass theorem and blow up analysis, we will investigate the compactness of the solutions to the fractional Yamabe problems. Meanwhile, under some geometric constraint conditions, we plan to prove the rigidity of the Riemann manifolds with nonnegative constant fractional scalar curvature by fractional Sobolev inequalities. The study of the problems which are related to the fractional scalar curvature provides an approach from using the fractional PDEs on Riemann manifolds, and it also reveals the deep connection between the fractional scalar curvature and fractional type positive mass theorem, and their important applications in conformal geometry.

分数阶数量曲率源自于分数阶Yamabe问题的研究，是非局部定义的几何量。本项目主要研究与分数阶数量曲率相关的分数阶Yamabe问题解集的紧性，同时考虑分数阶数量曲率为非负常数时黎曼流形的刚性。我们从经典的正质量定理的研究思路出发，尝试建立并证明分数阶正质量定理。通过考虑与分数阶Yamabe问题等价的边界退化椭圆偏微分方程狄利克雷到诺依曼边值问题，并利用分数阶正质量定理和爆破分析技术，从而得到分数阶Yamabe问题解集紧性的证明。同时，我们将利用分数阶Sobolev不等式，在一定的几何约束条件下，证明分数阶数量曲率为非负常数的黎曼流形的刚性。分数阶数量曲率的相关问题研究提供了黎曼流形上利用分数阶偏微分方程的方法，同时也揭示了分数阶数量曲率和分数阶正质量定理之间的深刻联系以及它们在共形几何中的应用价值。

本项目主要研究了分数阶数量曲率、sigma_2曲率等数量型曲率的几何和拓扑问题，包括与分数阶数量曲率相关的分数阶Yamabe问题解集的紧性、数量曲率为正的常数的Bach平坦的闭黎曼流形的曲率间隙现象、Brown-York拟局部质量对应的正质量定理与Besse猜想之间的关系以及关于sigma_2曲率的体积比较定理。我们得到的主要结果有：（1）分数阶Yamabe问题的解集在限制流形维数和分数阶指标的情况下紧性成立；（2）数量曲率为正的常数的Bach平坦闭黎曼流形的Weyl张量和迹为零的Ricci张量满足一定的有界性条件时，黎曼流形局部等距于标准球面；（3）我们找到了Brown-York拟局部质量对应的正质量定理与Besse猜想之间有意义的联系；（4）在严格稳定的正爱因斯坦度量的小邻域内，我们证明了关于sigma_2曲率的体积比较定理。