Metric Measure Spaces and Ricci Curvature Dynamics

Metric measure spaces extend the classical notion of smooth manifolds by equipping a set with both a distance function and a reference measure. This abstraction permits the study of spaces that may exhibit singularities or branching structures yet still support notions of curvature and dimension. Central to the modern theory is the synthetic interpretation of Ricci curvature bounds via optimal transport: by imposing a Curvature‐Dimension condition, one captures lower bounds on Ricci curvature and upper bounds on dimension without recourse to differentiable structure. The refinement to RCD(K,N) spaces further demands an infinitesimal Hilbertian property, ensuring that the energy functional behaves as in the smooth Riemannian setting. Together, these ideas have unlocked new insights into heat‐kernel behaviour, functional inequalities, geometric rigidity and stability under convergence. Applications range from the analysis of limit spaces arising in geometric flows to models of space–time singularities in gravitational theories, while the interplay with probability theory has deepened understanding of diffusion processes on nonsmooth backgrounds.

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Technical terms

Metric measure space: A set endowed with a distance function and a Borel measure, generalising Riemannian manifolds to potentially nonsmooth settings.

Ricci curvature bound: A synthetic condition expressing a lower bound on Ricci curvature via convexity properties of entropy along optimal‐transport geodesics.

Curvature‐Dimension condition (CD(K,N)): A requirement that a metric measure space behaves as if it has Ricci curvature ≥ K and dimension ≤ N, formulated through entropy convexity in the space of probability measures.

RCD(K,N) space: A metric measure space satisfying CD(K,N) together with an infinitesimal Hilbertian property, ensuring a quadratic energy functional akin to a Dirichlet form.

Wasserstein geodesic: A constant‐speed interpolation between probability measures minimising the cost of transport, used to probe curvature via displacement convexity.

Gromov–Hausdorff convergence: A notion of convergence for metric spaces based on mutual approximations, central to compactness and stability arguments in geometric analysis.

References

1. The globalization theorem for the Curvature-Dimension condition. Inventiones Mathematicae (2021).
2. Boundary regularity and stability for spaces with Ricci bounded below. Inventiones Mathematicae (2022).
3. Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds. Mathematische Annalen (2023).