Nonlinear diffusion equations and curvature conditions in metric measure spaces

TL;DR

This paper introduces new characterizations of the curvature-dimension condition in metric measure spaces using weighted action functionals and nonlinear diffusion semigroups, linking geometric and analytical perspectives.

Contribution

It provides novel approaches to characterize CD*(K,N) conditions through weighted action functionals and nonlinear diffusion semigroups, extending previous frameworks.

Findings

Weighted action functionals capture K-convexity of N-dimensional entropies.

Nonlinear diffusion semigroup approach is equivalent to heat flow under certain assumptions.

New characterizations unify geometric and diffusion-based perspectives on curvature conditions.

Abstract

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.