Characterizations of monotonicity of vector fields on metric measure spaces

Abstract

We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as "splitting theorem" and "volume cone implies metric cone theorem" in non-smooth context.In the past twenty years, the displacement convexity of functionals on Wasserstein space has been deeply studied, and it has applications in many fields such as differential equation theory, probability theory, differential and metric geometry (see [4] and [44] for an overview of related theories).One of the most interesting functionals is the Boltzmann entropy. Let (M, g, V g ) be a Riemannian manifold. The Boltzmann entropy Ent Vg (·) is defined byotherwise It is known from [45] that the convexity of Ent Vg (·) in Wasserstein space characterizes the lower Ricci curvature bound of M. It is proved by Erbar in [19] that the gradient flow of Ent Vg in Wasserstein space can be identified with the heat flow in the following sense: letH t (f V g ) be the Wasserstein gradient flow of Ent Vg starting from f V g ∈ P 2 (M), H t (f ) be the solution of the heat equation with initial datumMoreover, we have the following well-known theorem. Theorem 1.1 (Von Renesse-Sturm [45], Erbar [19]). Let (M, g, V g ) be a Riemannian manifold. Then the following characterizations are equivalent.1) The Ricci curvature of M is uniformly bounded from below by a constant K ∈ R.2) The entropy Ent Vg (·) is K-convex in Wasserstein space.3) For any probability measure µ, there exists a unique EVI K -gradient flow of Ent Vg in Wasserstein space starting from µ.4) The exponential contraction of the heat flows in Wasserstein distanceholds for any two heat flows µ i t := H t (f i )V g , i = 1, 2. 5)There exits a heat kernel ρ t (x, dz)V g (z) =H t (δ x ) for any x ∈ X, such that the exponential contraction of heat kernels in Wasserstein distanceholds for any x, y ∈ X and t > 0. 6) The gradient estimate of heat flow|DH t (f )| 2 (x) ≤ e −2Kt H t (|Df | 2 )(x), V g − a.e. x ∈ Xholds for any f ∈ W 1,2 (M).

Rigidity of some functional inequalities on RCD spaces

“…We know that |∇u| is a positive constant and u is Lipschitz. By [30,Theorem 4.4] (or [34,Theorem 3.16]), we know that the gradient flow (F t ) t≥0 of u, which is also the regular Lagrangian flow associated with −∇u in the sense of Ambrosio-Trevisan [13, §8], satisfies the following equality (see also [28])…”