Geometry of measure concentration and curvature

我们详细研究了格罗莫夫提出的度量度量空间的几何理论，并为此写了一本书。证明了如果度量空间序列关于可观测距离收敛到度量度量空间，则曲率-维条件是稳定的。作为应用，我们给出了具有非负Ricci曲率的闭黎曼流形上拉普拉斯算子的第k个特征值与第一个特征值之比的估计，其中该估计仅依赖于k。Gromov定义了度量度量空间的一个自然紧化。我们对其进行了深入的考虑，并在其上引入了一种新的度量结构。我们利用我们的度量结构证明了欧氏空间中半径为n的平方根的n维球面当n趋于无穷大时收敛到无限维高斯空间。

We studied the details of a geometric theory of metric measure spaces due to Gromov and wrote a book for it. We proved that if a sequence of metric measure spaces converges to a metric measure space with respect to the observable distance, then the curvature-dimension condition is stable. As an application, we give an estimate of the ratio of the k-th eigenvalue and the first eigenvalue of the Laplacian on a closed Riemannian manifold with nonnegative Ricci curvature, where the estimate depends only on k. Gromov defined a natural compactification of the space of metric measure spaces with the observable distance. We deeply considered it and introduce a new metric structure on it. We apply our metric structure to prove that an n-dimensional sphere of radius square root of n in a Euclidean space converges to an infinite-dimensional Gaussian space as n tends to infinity.