Calculus of Variations and Geometric Measure Theory

R. Buzano

Ricci flow coupled with harmonic map flow

created by muller on 12 Jun 2018
modified on 13 Jun 2022


Published Paper

Inserted: 12 jun 2018
Last Updated: 13 jun 2022

Journal: Annales Scientifiques de l'École Normale Supérieure
Volume: 45
Year: 2012
Doi: 10.24033/asens.2161

ArXiv: 0912.2907 PDF

Published under previous name Reto Müller (please cite as Müller). Name changed here in order to import to author page correctly.


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 Falpha 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 Walpha 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.