Ricci Flow : The Foundations via Optimal Transportation

Peter Topping (1)

1 Mathematics Institute

We give an introduction to some of the main concepts of Ricci flow, including L-length, monotonic quantities, Harnack estimates and log-Sobolev inequalities, and explain them from the point of view of Optimal Transportation. Most of the explanation is based around the construction of an $n+1$-dimensional Ricci flow on "space-time" -- a so-called Canonical Soliton.

Mots clés: Ricci flow; Optimal Transportation; Entropy; Log-Sobolev inequalities; Harnack estimates; Curvature; Ricci soliton; Canonical soliton; Wasserstein distance; L-optimal transportation; L-Wasserstein distance

Domaines: Analyse classique [math.CA]; Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Peter Topping. Ricci Flow : The Foundations via Optimal Transportation. 2009. ⟨hal-00529642⟩