Stochastic Particle Approximations for the Ricci Flow on Surfaces and the Yamabe Flow

Abstract

We present stochastic particle approximations for the normalized Ricci flow on surfaces and for the non-normalized Yamabe flow on manifolds of arbitrary dimension.Keywords: Ricci flow on surfaces, Yamabe flow, stochastic interacting particle system. AMS subject classification: 53C44, 82C22. Ricci and Yamabe flowThanks to Perelman's seminal work on the geometrization and hence the Poincaré conjecture [19,20,21] the Ricci flow has attracted worldwide attention and led to many new developments (see e.g. [17]). The importance of this subject has been underlined by the award of the Fields medal to Perelman. There is now a strong interest in understanding the microscopic structure of the Ricci flow.The evolution of a Riemannian metric g = g t on a connected d-dimensional closed manifold M under the (normalized) Ricci flow is described by the partial differential equationHere Ric is the Ricci curvature andR the average scalar curvature of M , i.e.R := 1 vol(M ) M R, where R is the scalar curvature (all quantities taken with respect to g t ).In dimension d = 2 we have Ric = 1 2 Rg, so that in this case (1) 2. As t → ∞, g t converges in any C k -norm to a smooth metric g ∞ of constant curvature.