Optimal Transport Theory and Wasserstein Geometry for Gradient Flows in Metric Spaces

Mathematical Applications of Science Fiction We present a rigorous derivation of the existence and unique-ness of gradient flows for semi-convex functionals on met-ric measure spaces satisfying the Riemannian Curvature-Dimension condition RCD∗(K, N ). By constructing amotivic transport operator within the Wasserstein spaceW2(X), we extend the Jordan-Kinderlehrer-Otto (JKO)scheme to non-smooth structures exhibiting perfectoid sin-gularities. We establish the Evolution Variational Inequal-ity (EVI) for the generalized entropy functional, provingK-contraction of the flow. Furthermore, we analyze thespectral convergence of the discrete-time approximationsusing a Hilbert-Schmidt embedding of the transport plans,providing a robust framework for dissipative dynamics incrystalline geometries. The analysis avoids standard co-homological obstructions by utilizing a derived symplecticformalism on the tangent bundle of the probability space.