Nonlinear diffusion equations and curvature conditions in metric measure
spaces

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, 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
CD*(K,N) condition of Bacher-Sturm.