Ricci flow coupled with harmonic map flow

TL;DR

This paper introduces a coupled Ricci and harmonic map flow that exhibits improved singularity behavior and monotonicity properties, extending Perelman's techniques to a new geometric evolution system.

Contribution

It presents a novel coupled flow system combining Ricci and harmonic map flows, with new energy estimates and monotonicity formulas that prevent certain singularities.

Findings

Coupled flow can be less singular than individual flows.

Energy concentration can be ruled out without curvature assumptions.

Monotonicity of entropy and volume functionals analogous to Perelman's results.

Abstract

We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of phi a-priori - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for alpha = 0. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.