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The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces
About this Title
Karl-Theodor Sturm
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 290, Number 1443
ISBNs: 978-1-4704-6696-1 (print); 978-1-4704-7629-8 (online)
DOI: https://doi.org/10.1090/memo/1443
Published electronically: October 24, 2023
Table of Contents
Chapters
- Introduction and Main Results at a Glance
- 1. The Metric Space $(\mathbb {X}_p, \Delta \!\!\!\!\Delta _p)$
- 2. The Topology of $(\mathbb {X}_p, \Delta \!\!\!\!\Delta _p)$
- 3. Geodesics in $(\mathbb {X}_p, \Delta \!\!\!\!\Delta _p)$
- 4. Cone Structure and Curvature Bounds for $(\mathbb {X}, \Delta \!\!\!\!\Delta )$
- 5. The Space $\mathbb {Y}$ of Gauged Measure Spaces
- 6. The Space $\mathbb {Y}$ as a Riemannian Orbifold
- 7. Semiconvex Functions on $\mathbb {Y}$ and their Gradients
- 8. The $\mathcal {F}$-Functional
- 9. Addendum: The $L^{p,q}$-Distortion Distance
Abstract
Equipped with the $L^{2,q}$-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.- Romain Abraham, Jean-François Delmas, and Patrick Hoscheit, A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces, Electron. J. Probab. 18 (2013), no. 14, 21. MR 3035742, DOI 10.1214/EJP.v18-2116
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