Ricci flow coupled with harmonic map flow (Q2885456)

Two important geometric flows are coupled in this paper, the Ricci flow \(g_t=-2 Rc\) for Riemannian metrics \(g\) on a smooth manifold \(M\) and the harmonic map heat flow \(\phi_t=\tau_g\phi\) for mappings \(\phi:M\to N\) between smooth manifolds, where here \(\tau_g\phi\) denotes the tension field with respect to \(g\), i.e., the part of \(\Delta_g\phi\) tangential to \(N\). For the coupled flow, the vector field \(\phi\) is used to modify the metric, \(g_t=-2Rc+2\alpha\,\nabla\phi\otimes\nabla\phi\), and the harmonic map heat flow is taken with respect to the evolving metric, \(\phi_t=\tau_g\phi\). The parameter \(\alpha>0\) can still be chosen as a constant or a function of \(t\), and a large \(\alpha\) will make solutions more regular. Special cases are List's flow (\(N\subseteq R\)) (motivated from general relativity, it later turned out to be a special case of Ricci flow) and also a flow considered by Williams (\(N\subseteq SL(k,R)/SO(k)\)), see [\textit{B. List}, ``Evolution of an extended Ricci flow system'', Commun.\ Anal.\ Geom.\ 16, No.\ 5, 1007--1048 (2008; Zbl 1166.53044)] and [\textit{M. Williams}, ``Results on coupled Ricci and harmonic map flows,'' \url{arXiv:1012.0291}].NEWLINENEWLINEThe paper discusses quite a number of aspects of the coupled flow. It starts with explicit examples, soliton solutions, and a volume-preserving version. For constant \(\alpha\), variants of Perelman's techniques for the Ricci flow are developed. It is proven that the flow can be interpreted as a gradient flow for an energy functional \(F_\alpha(g,\phi,f)\) modified by a familiy of diffeomorphisms generated by \(\nabla f\), and monotonicity of \(F_\alpha\) can be discussed similarly to the Ricci flow. DeTurck's trick is applied to prove short-time existence. Existence or non-existence of cerain types of singularities is discussed. A uniform bound on the Riemannian curvature implies long-time existence. Perelman's shrinker entropy is generalized to the coupled flow. As a consequence, nontrivial breathers cannot exist if \(\alpha\) is non-increasing. Finally, a local non-collapsing theorem is proven.