Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case

By Luigi Ambrosio, Nicola Gigli, Giuseppe Savar'e. 2012

47 citations recorded by the source corpus.

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L 2-gradient flow of a suitably defined “Dirichlet energy” and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier’s Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ac5f5d4c6bb8ff6ac67ce5f6970bb4e0944ff819

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Citations in the corpus

Metric viscosity solutions of Hamilton–Jacobi equations depending on local slopes (2015, background): “As regards equations in metric spaces, Hamilton–Jacobi semigroups, and pointwise differential inequalities involving local slopes satisfied by them have been investigated in [3, 4, 5, 6, 42, 43, 53], mostly in connection with applications to various functional inequalities.”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “A second step (first developed in [48] and then generalized in [5,6]) consists in providing existence and uniqueness conditions for the gradient flow of the entropy, w.”
On the differential structure of metric measure spaces and applications (2012, background): “We refer to [24] for an overview on the subject and detailed bibliography (see also [7] for recent progresses).”