Flowing maps to minimal surfaces: Existence and uniqueness of solutions

TL;DR

This paper investigates a geometric flow that transforms initial data into branched minimal immersions, proving existence and uniqueness of solutions under certain conditions, thereby advancing understanding of minimal surface mappings.

Contribution

It establishes the existence and uniqueness of weak solutions to the flow, given boundedness in moduli space and non-increasing energy, extending previous work on flow convergence.

Findings

Existence of weak solutions as long as the metric remains in a bounded moduli space region.

Uniqueness of solutions among all weak solutions with non-increasing energy.

Flow does not collapse a closed geodesic in finite time under the given conditions.

Abstract

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.