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 Bacher-Sturm.
Introduction
Contraction and Convexity via Hamiltonian Estimates: An Heuristic Argument
Part I. Nonlinear Diffusion Equations and Their Linearization in Dirichlet
Spaces: Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion
Backward and Forward Linearizations of Nonlinear Diffusion
Part II. Continuity Equation and Curvature Conditions in Metric Measure
Spaces: Preliminaries
Absolutely Continuous Curves in Wasserstein Spaces and Continuity
Inequalities in a Metric Setting
Weighted Energy Functionals along Absolutely Continuous Curves
Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger
Energies
The $\mathrm{RCDS}^{*}(K, N)$ Condition and Its Characterizations through
Weighted Convexity and Evolution Variational Inequalities
Part III. Bakry-Emery Condition and Nonlinear Diffusion: The Bakry-Emery
Condition
Nonlinear Diffusion Equations and Action Estimates
The Equivalence Between $\mathrm{BE}(K, N)$ and $\mathrm{RCDS}^{*}(K, N)$
Bibliography.
Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy.
Andrea Mondino, University of Warwick, Coventry, United Kingdom.
Giuseppe Savare, Universita di Pavia, Italy.
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