RCD空间的性质研究及其应用

RCD*(K,N) spaces are metric measure spaces with a suitable notion of ‘Ricci curvature bounded from below by K and dimension bounded from above by N’. Riemannian manifolds with Ricci curvature lower bound and their measured Gromov-Hausdorff limits are examples of RCD spaces. Recently, the research on RCD spaces has become an important branch in geometric analysis. In this project, we will study some properties and applications of RCD spaces. More precisely, we will consider the following three topics. Recently it is proved by some scholars that almost every point on an RCD space has a unique tangent cone which is isomorphic to an Euclidean space, but it is still unclear whether the dimensions of these tangent cones are independent of the points. Thus the first topic in this project is to study the problem that whether almost all points on an RCD space have the same Euclidean space as tangent cones. Secondly, we will study the properties of noncompact RCD*(0,N) spaces with linear volume growth and try to prove that on such spaces the diameter of the level sets of a Busemann function grows sublinearly. Thirdly, we will try to apply the RCD theory to study the existence of non-constant harmonic functions with polynomial growth on noncompact manifolds with nonnegative Ricci curvature. The last two topics are based on the previous works of the proposer.

RCD*(K,N)空间是一种“Ricci曲率下界为K，维数上界为N”的度量测度空间，Ricci曲率有下界的黎曼流形以及其Gromov-Hausdorff极限都是RCD空间的基本例子。近年来对RCD空间的研究已经成为几何分析的重要分支。本项目将会围绕RCD空间的性质和应用展开研究，研究内容分三方面。一些学者的研究表明了RCD空间上几乎处处点的切锥都等距于欧氏空间，但这些切锥的维数是否与点的选取无关则是未知的。因此，本项目第一个研究内容拟围绕RCD空间上点的切锥维数的几乎处处唯一性问题展开。其次，本项目拟研究有线性体积增长的非紧RCD*(0,N)空间的性质，并证明在这种空间上Busemann函数的水平集的直径是次线性增长的。最后，本项目希望应用RCD理论来研究有非负Ricci曲率的非紧流形上多项式增长调和函数的存在性。其中，后两个研究内容是在申请人已有工作的基础上更为深入的展开。

RCD*(K,N)空间是一种“Ricci曲率下界为K，维数上界为N”的度量测度空间，Ricci曲率有下界的黎曼流形以及其Gromov-Hausdorff极限都是RCD空间的基本例子。近年来对RCD空间的研究已经成为几何分析的重要分支。本项目主要围绕RCD空间的性质和应用展开研究。其中，主要研究内容包括：有线性体积增长的非紧RCD*(0,N)空间的性质；RCD空间上“几乎体积刚性”类型定理的研究；Ricci曲率非负的非紧流形上多项式增长调和函数的性质以及多项式增长调和函数的存在性问题，等等。在本基金的支持下，我们最终得到的重要结果包括：对有线性体积增长的非紧RCD(0,N)空间的系统研究；RCD空间上的“几乎体积刚性”类型定理；在一定条件下，Ricci曲率非负的非紧流形上多项式增长调和函数空间维数的下界和渐近估计，等等。这些结果丰富和完善了Ricci曲率有下界的流形和度量测度空间的理论。

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