An almost rigidity theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth
An almost rigidity theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth
复制标题
几乎刚性定理及其在体积线性增长的非紧RCD(0,N)空间中的应用
DOI：
10.1142/s0219199718500761
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发表时间：
2020
影响因子：
1.6
通讯作者：
Xian-Tao Huang
中科院分区：
文献类型：
--
作者：
Xian-Tao Huang
本文的主要研究成果由两部分组成。首先，我们得到了一个几乎刚性定理：在RCD(0，N)空间上，当距离函数的两个水平集之间的区域相对于柱面的体积几乎最大时，则该部分作为度量空间接近于柱面。其次，我们利用这个几乎刚性定理研究了具有线性体积增长的非紧RCD(0，N)空间。更确切地说，我们得到了测地球面直径的次线性增长，并在这样的RCD(0，N)空间上研究了具有多项式增长的非常数调和函数的不存在性问题。
The main results of this paper consist of two parts. First, we obtain an almost rigidity theorem which roughly says that on an RCD(0,N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Second, we apply this almost rigidity theorem to study noncompact RCD(0,N) spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence problem of nonconstant harmonic functions with polynomial growth on such RCD(0,N) spaces.