RCD spaces: Splitting theorems and applications.

About

RCD(K,N) spaces are a synthetic generalization of manifolds with Ricci curvature bounded below and dimension bounded above. They are stable under measured Gromov - Hausdorff convergence, and so are also a generalization of Ricci limit spaces. Several now classical results such as Cheeger-Gromoll's Splitting Theorem and Cheeger-Colding's Almost Splitting Theorem have been generalized to this setting.

In this minischool we will review some of the new proofs and how either the theorems or the ideas behind the proofs have been used to prove other results such as the volume cone implies metric cone (Gigli-De Phillips); stability of the torus (Mondello - Mondino - Perales) and; volume entropy rigidity (Connell - Dai - Nuñez Zimbrón - Perales -Suárez Serrato- Wei), etc.

Schedule

(Mexico City's Time Zone, i.e CST: UTC-06:00)

09:30-10:30	Cheeger-Gromoll splitting theorem Mauricio Che Moguel, University of Durham We will review the proof of the classical Cheeger-Gromoll splitting theorem for riemannian manifolds and its generalization for Alexandrov spaces.
10:40-11:40	Introduction to RCD spaces Jaime Santos Rodríguez, MPIM Bonn
11:50-12:50	A maximal diameter theorem for metric measure spaces with positive Ricci curvature bounded below Christian Ketterer, University of Cologne We will survey rigidity theorems under positive lower Ricci curvature bounds in the context of metric measure spaces. This will include a maximal diameter theorem and some of its consequences like the sphere theorem or Obata's theorem. I outline key steps and ideas of the proofs. In particular I explain how Euclidean metric cones area useful tool therein.

09:30-10:30

Cheeger-Gromoll splitting theorem

Mauricio Che Moguel, University of Durham

We will review the proof of the classical Cheeger-Gromoll splitting theorem for riemannian manifolds and its generalization for Alexandrov spaces.

10:40-11:40

Introduction to RCD spaces

Jaime Santos Rodríguez, MPIM Bonn

11:50-12:50

A maximal diameter theorem for metric measure spaces
with positive Ricci curvature bounded below

Christian Ketterer, University of Cologne

We will survey rigidity theorems under positive lower Ricci curvature bounds in the context of metric measure spaces. This will include a maximal diameter theorem and some of its consequences like the sphere theorem or Obata's theorem. I outline key steps and ideas of the proofs. In particular I explain how Euclidean metric cones area useful tool therein.

9:30-10:30	Splitting theorem of Gigli Enrico Pasqualetto, Scuola Normale Superiore We will review the splitting theorem that appears in https://arxiv.org/pdf/1302.5555.pdf
10:40-11:40	Mondino-Naber almost splitting theorem Daniele Semola, University of Oxford We will review the almost splitting theorem that appears in https://arxiv.org/pdf/1405.2222.pdf
11:50-12:50	Recognizing the flat torus among \(\textrm{RCD}^*(0,N)\) spaces via the study of the first cohomology group Chiara Rigoni, University of Vienna We will review the paper with the same title as this talk and put emphasis on describing regular lagrangian flows https://arxiv.org/abs/1705.04466

9:30-10:30	Gigli-De Philippis Volume cone to metric cone David Tewodrose, Université Libre de Bruxelles We will review the main result in this paper: https://arxiv.org/pdf/1512.03113.pdf
10:40-11:40	Torus stability for RCD spaces of Mondello - Mondino - Perales Ilaria Mondello, University of Paris-Est Créteil We will review the stability result that appears in https://arxiv.org/pdf/2104.06208.pdf
11:50-12:50	Volume Entropy Rigidity of Connell - Dai - Núñez Zimbrón - Perales -Suárez Serrato - Wei Jesús Núñez Zimbrón, CIMAT We will review the rigidity theorem that appears in https://arxiv.org/pdf/1809.06909.pdf