Limiting Behaviour of the Teichm\"uller Harmonic Map Flow

In this paper we study the Teichm\"uller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where $u$ maps from a fixed closed surface $M$ with metric $g$ to a general target manifold $N$. It arises naturally as a gradient flow for the Dirichlet energy functional viewed as acting on equivalence classes of such pairs, obtained from the invariance under diffeomorphisms and conformal changes of the domain metric. In the construction of a suitable inner product for the gradient flow a choice of relative weight of the map tangent directions and metric tangent directions is made, which manifests itself in the appearance of a coupling constant $\eta$ in the flow equations. We study limits of the flow as $\eta$ approaches 0, corresponding to slowing down the evolution of the metric. We first show that given a smooth harmonic map flow on a fixed time interval, the Teichm\"uller harmonic map flows starting at the same initial data converge uniformly to the underlying harmonic map flow when $\eta \downarrow 0$. Next we consider a rescaling of time, which increases the speed of the map evolution while evolving the metric at a constant rate. We show that under appropriate topological assumptions, in the limit the rescaled flows converge to a unique flow through harmonic maps with the metric evolving in the direction of the real part of the Hopf differential.