具有非负数量曲率带边流形上的一些几何分析问题研究

在本项目中，申请人计划研究具有非负数量曲率带边流形上的一些几何分析问题。研究这些问题，既有助于理解数量曲率有下界流形上的几何，也有助于我们更好地认识广义相对论中的正质量定理和拟局部质量。本项目的主要研究目标为：.(1)对闭流形(M,g)，找到一个依赖维数,以及(M,g)上具体几何量的h，使得当H>h时，(M,g,H)不存在数量曲率非负的填充（填充的定义详见正文第三页）。.(2)给定闭流形(M,g)，其Λ_+不变量为满足下述条件的H在(M,g)上积分的上确界，这里H为正且(M,g,H)允许一个数量曲率非负的填充。我们想研究Λ_+(M,g)关于度量g的C^0连续性以及利用Λ_+不变量来证明渐近平坦流形上的正质量定理蕴含渐近双曲流形上的正质量定理。.(3)试图证明积分条件下的黎曼多面体比较定理。.上述研究内容与Gromov最近提出的数量曲率填充问题以及多面体比较问题密切相关。

In this program, we are going to study some geometric problems on Riemannian manifolds with boundary with nonnegative scalar curvature.The study of these problems can help us understand not only the geometry of manifolds with scalar curvature bounded from blow, but also the positive mass theorem and the quasi-local mass in general relativity. The main goals of this program are the following:.(1) For any closed Riemannian manifold (M,g),find a constant h depending on its dimension and the geometric quantities on (M,g) such that when H>h, (M,g,H) admits no fill-in of nonnegative scalar curvatures (for the definition of fill-in, see page 3 in the text)..(2)Given a closed Riemannian manifold (M,g), its Λ_+ invariant is defined to be the supremum of the integral of H on (M,g) where H satisfies H>0 and (M,g,H) admits a fill-in of nonnegative scalar curvature. We want to study the continuity of Λ_+(M,g) with respect to the C^0 topology of g and use Λ_+ invariant to prove the positive mass theorem for asymptotically flat manifolds implies the positive mass theorem for asymptotically hyperbolic manifolds..(3) Try to prove the Riemannian Polyhedra comparison theorem under the integral condition..These goals are both closely related to some problems related to fill-ins and polyhedra comparison proposed by Gromov recently.