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Softcover ISBN:  9781470439132 
Product Code:  MEMO/262/1270 
List Price:  $81.00 
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Product Code:  MEMO/262/1270.E 
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AMS Member Price:  $48.60 
Softcover ISBN:  9781470439132 
eBook ISBN:  9781470455132 
Product Code:  MEMO/262/1270.B 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 262; 2019; 121 ppMSC: Primary 49; 35; Secondary 58; 31
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces \((X,\mathsf d,\mathfrak m)\).
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 \(\mathrm {CD}^{*}(K,N)\) condition of BacherSturm.

Table of Contents

Chapters

1. Introduction

2. Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument

1. Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces

3. Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion

4. Backward and Forward Linearizations of Nonlinear Diffusion

2. Continuity Equation and Curvature Conditions in Metric Measure Spaces

5. Preliminaries

6. Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting

7. Weighted Energy Functionals along Absolutely Continuous Curves

8. Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies

9. The $\ensuremath {\mathrm {RCD}^*(K,N)}$ Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities

3. BakryÉmery Condition and Nonlinear Diffusion

10. The BakryÉmery Condition

11. Nonlinear Diffusion Equations and Action Estimates

12. The Equivalence Between $\mathrm {BE}(K,N)$ and $\ensuremath {\mathrm {RCD}^*(K,N)}$


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The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces \((X,\mathsf d,\mathfrak m)\).
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 \(\mathrm {CD}^{*}(K,N)\) condition of BacherSturm.

Chapters

1. Introduction

2. Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument

1. Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces

3. Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion

4. Backward and Forward Linearizations of Nonlinear Diffusion

2. Continuity Equation and Curvature Conditions in Metric Measure Spaces

5. Preliminaries

6. Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting

7. Weighted Energy Functionals along Absolutely Continuous Curves

8. Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies

9. The $\ensuremath {\mathrm {RCD}^*(K,N)}$ Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities

3. BakryÉmery Condition and Nonlinear Diffusion

10. The BakryÉmery Condition

11. Nonlinear Diffusion Equations and Action Estimates

12. The Equivalence Between $\mathrm {BE}(K,N)$ and $\ensuremath {\mathrm {RCD}^*(K,N)}$