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.