Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

About this Title

Luigi Ambrosio, Classe di Scienze, Scuola Normale Superiore, Pisa. Piazza dei Cavalieri 7, Pisa, Italy, Andrea Mondino, Mathematics Institute, University of Warwick. Coventry CV4 7AL, UK and Giuseppe Savaré, Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata 1, Pavia, Italy

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 262, Number 1270
ISBNs: 978-1-4704-3913-2 (print); 978-1-4704-5513-2 (online)
Published electronically: December 18, 2019
Keywords: Optimal transport, Ricci curvature, Metric measure spaces, Bakry-Émery tensor, Nonlinear diffusion, Displacement convexity, Nonsmooth Riemannian geometry
MSC: Primary 49J52, 49Q20, 35K55; Secondary 58J35, 35K90, 31C25, 58B20

View full volume PDF

View other years and numbers:

Table of Contents


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

3. Bakry-Émery Condition and Nonlinear Diffusion

  • 10. The Bakry-Émery Condition
  • 11. Nonlinear Diffusion Equations and Action Estimates
  • 12. The Equivalence Between $\mathrm {BE}(K,N)$ and $\mathrm {RCD}^*(K,N)$


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, our 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, our 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 Bacher-Sturm.

References [Enhancements On Off] (What's this?)