非负数量曲率填充问题的研究

一组(推广的)Bartnik数据(M,g,h)由一个可定向、零配边的闭黎曼流形(M,g)及其上一个光滑函数h构成。(M,g,h)的一个非负数量曲率填充是指一个数量曲率非负的紧致带边流形，其边界与(M,g)等距同构且关于外法向的平均曲率为h。研究非负数量曲率填充问题，既有助于刻画数量曲率非负的紧致带边流形的边界行为，也有助于理解广义相对论中的拟局部质量。定义一个不变量Λ_+(M,g)为：所有正的、(M,g,h)允许非负数量曲率填充的h中，h在(M,g)上积分的上确界。本项目主要的研究目标是：.(1)当M是拓扑球面时，证明Λ_+(M,g)的有限性及关于g的C^0拓扑的连续性；.(2)借助Λ_+不变量定义推广的Brown-York质量，通过研究Λ_+不变量关于拓扑球面在标准度量处的展开，得出推广的Brown-York质量的大小尺度极限。.其中(1)与Gromov最近提出的两个猜想密切相关。

A triple of (generalized) Bartnik data (M,g,h) consists of an orientable closed null-cobordant Riemannian manifold (M,g) and a given smooth function h on it. A fill-in of nonnegative scalar curvature of the Bartnik data (M,g,h) is a compact Riemannian manifold with nonnegative scalar curvature whose boundary is isometric to (M,g) and has mean curvature h with respect to the outward normal. The study of fill-in of nonnegative scalar curvature helps us to describe the boundary behavior of a compact Riemannian manifold with boundary and nonnegative scalar curvature and understand the quasi-local mass in general relativity. Define Λ_+(M,g) to be the supremum of the integral of h on (M,g) among all positive h that (M,g,h) admits a fill-in of nonnegative scalar curvature. The goals of this program are the following..(1) When M is a topological sphere, prove the finiteness of Λ_+(M,g) and the continuity of Λ_+(M,g) with respect to the C^0 topology of g..(2) Define a generalized Brown-York mass by the Λ_+ invariant and obtain the large scale limit and small scale limit of the generalized Brown-York mass via the study of the expansion of Λ_+ at the standard metric on the sphere..The first goal is closely related to two conjectures raised by Gromov recently.