Nicola Gigli: Introduction to the Riemannian Curvature Dimension condition

5th September, 2022

This is a 22-lecture course, with each lecture being about 90 minutes, given by Nicola Gigli. Note that the 14th and 18th lectures were not recorded.

Created by Professor Nicola Gigli, the aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of Jordan-Kinderlehrer-Otto we shall see that such heat flow can be equivalently characterized as gradient flow of the Cheeger-Dirichlet energy on L2 and as gradient flow of the Boltzmann-Shannon entropy w.r.t. the optimal transportation metric W2. This provides a crucial link between the Lott-Villani-Sturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of ‘infinitesimally Hilbertian’ spaces as those metric measure structures for which W1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds. 

We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.

We shall finally see how these are linked to the lower Ricci curvature bound – most notably we shall prove the Laplacian comparison theorem – and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth noticing that such statement gives new information – compared to those available through Cheeger-Colding’s theory of Ricci-limit spaces – even about the structure of smooth Riemannian manifolds.

Prerequisites: some familiarity with Riemannian geometry and optimal transport theory in the case cost=distance2 is preferred, but not required: I shall provide the necessary background when needed.

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
  9. Lecture 9
  10. Lecture 10
  11. Lecture 11
  12. Lecture 12
  13. Lecture 13
  14. Lecture 14
  15. Lecture 15
  16. Lecture 16
  17. Lecture 17
  18. Lecture 18
  19. Lecture 19
  20. Lecture 20
  21. Lecture 21
  22. Lecture 22

These videos were produced by the Fields Institute, as a graduate course (link to course page) belonging to the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry.