The paper concerns dimension-curvature conditions and contraction properties of the heat semigroup \(P_{t}\) on Riemannian manifolds with respect to the Wasserstein distance \(W_{2}\) between probability measures. One of the first results in this direction is by M.-K. von Renesse and K.-T. Sturm [Commun. Pure Appl. Math. 58, No. 7, 923–940 (2005; Zbl 1078.53028)], who showed that on a complete Riemannian manifold the Ricci curvature being bounded by \(K\) is equivalent to \(W_{2}^{2}\left( P_{t}fdx, P_{t}gdx \right)\leqslant e^{-2Kt}W_{2}^{2}\left( fdx, gdx \right)\), where \(dx\) is the Riemannian volume element, and \(f, g\) are probability densities.
The current paper deals with one of the directions of this equivalence proving that a curvature-dimension inequality implies a contraction property involving both a curvature bound and a dimension for a weighted compact Riemannian manifold. The author uses the Benamou-Brenier dynamical formulation of the Wasserstein distance to get a dimensional estimate on the Hodge-de Rham semigroup. This is the semigroup with the generator being the Hodge-de Rham operator on forms, and the estimates rely on the Bochner-Lichnerowicz-Weitzenböck identity for \(1\)-forms.