A stochastic target approach to Ricci flow on surfaces

Abstract

We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi’s representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation.

Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^{k}$-norm. In the case of $C^{0}$ and $C^{1}$-convergence, we achieve this by coupling two particles. To get $C^{2}$-convergence (in particular, convergence of the curvature), we use a coupling of three particles. This triple coupling is developed here only for the case of constant curvature metrics on surfaces, though we suspect that some variants of this idea are applicable in other situations and therefore be of independent interest. Finally, for $k\ge3$, the $C^{k}$-convergence follows relatively easily using induction and coupling of two particles.

None of these techniques appear in the Ricci flow literature and thus provide an alternative approach to the field.