The Space Of Spaces Curvature Bounds And Gradient Flows On The Space Of Metric Measure Spaces
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The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces
Author: Karl-Theodor Sturm
language: en
Publisher: American Mathematical Society
Release Date: 2023-11-27
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Alexandrov Geometry
Author: Stephanie Alexander
language: en
Publisher: American Mathematical Society
Release Date: 2024-05-24
Alexandrov spaces are defined via axioms similar to those of the Euclid axioms but where certain equalities are replaced with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above (CBA) and curvature bounded below (CBB). Even though the definitions of the two classes of spaces are similar, their properties and known applications are quite different. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering Alexandrov spaces with CBA and with CBB simultaneously. The book also includes an extensive problem list with solutions indicated for every problem.
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
Author: Luigi Ambrosio
language: en
Publisher: American Mathematical Soc.
Release Date: 2020-02-13
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.