The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces

IF 2 4区数学 Q1 MATHEMATICS Memoirs of the American Mathematical Society Pub Date : 2023-10-01 DOI:10.1090/memo/1443

Karl-Theodor Sturm

引用次数: 78

Abstract

Equipped with the L 2 , q L^{2,q} -distortion distance

\DD _{2,q}

, the space

\XX _{2q}

of all metric measure spaces

(X,\d ,\m )

is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on

\ol \XX _{2q}

are presented.

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空间的空间:度量空间空间上的曲率边界和梯度流

利用l2,q L^{2,q} -畸变距离\DD _{2,q}，证明了所有度量测量空间(X，\d，\m)的空间\XX _{2q}在Alexandrov意义上具有非负曲率。详细描述了测地线和切线空间。此外，还给出了半凸泛函的类及其在\ol \XX _{2q}上的梯度流。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Memoirs of the American Mathematical Society 数学-数学

CiteScore

3.50

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.

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