Optimal Transport Regularity and the Geometry of Ricci Curvature Bounds in Metric Measure Spaces

This monograph investigates the structural interplay between the regularity of optimal transport maps and synthetic Ricci curvature bounds in metric measure spaces (X, d, m). We analyze the foundational equivalence between the curvature-dimension condition CD(K, N) and the displacement convexity of the Boltzmann-Shannon entropy on the Wasserstein space P2(X). Particular attention is devoted to the Riemannian Curvature-Dimension condition RCD*(K, N), which enforces infinitesimal Hilbertianity, thereby ruling out Finslerian geometries and ensuring the linearity of the heat flow. We discuss the recent breakthroughs concerning the partial regularity of optimal maps in the non-smooth setting, extending the classical Monge-Ampère regularity theory of Caffarelli, Ma, Trudinger, and Wang to the metric realm. Furthermore, we examine the geometric rigidity consequences of these transport estimates, including the splitting theorem and the rectification of the singular set. The analysis relies on advanced techniques from non-smooth analysis, geometric measure theory, and the calculus of variations.