  We study the new geometric flow that was introduced in [11] that evolves a
pair of map and (domain) metric in such a way that it changes appropriate
initial data into branched minimal immersions. In the present paper we focus on
the existence theory as well as the issue of uniqueness of solutions. We
establish that a (weak) solution exists for as long as the metrics remain in a
bounded region of moduli space, i.e. as long as the flow does not collapse a
closed geodesic in the domain manifold to a point. Furthermore, we prove that
this solution is unique in the class of all weak solutions with non-increasing
energy. This work complements the paper [11] of Topping and the author where
the flow was introduced and its asymptotic convergence to branched minimal
immersions is discussed.
