Title: Limiting Behaviour of the Teichmüller Harmonic Map Flow,
Abstract: In this paper we study the Teichmüller 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 Teichmüller 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.