Analysis of gradient flows relating to minimal surfaces

Thesis

In this thesis, I study certain geometric gradient flow equations, some old, some new, which all seek to construct minimal surfaces.

The first equation I study is the half-harmonic map flow, also known as the Plateau flow, which is an evolution equation for maps from the circle S1 into a smooth closed submanifold N of Rn. This is a gradient flow for the half-energy functional, introduced by Da Lio and Rivière, the critical points of which are half-harmonic maps. These are of geometric interest since the harmonic extension of a half-harmonic map to the unit disc parametrises a free boundary minimal surface. This flow equation has been studied by several authors, in particular by Wettstein and Struwe and it is two questions of Struwe which I will answer in this part of the thesis. The first of these is to prove uniqueness of weak solutions along which the energy is non-increasing, improving upon the existing uniqueness result proved by Struwe and bringing the theory in line with what is known about the harmonic map flow. The second question concerns the relation of half-harmonic maps with the classical Plateau problem when the target N is a closed curve Γ. In particular, I prove the monotonicity of half-harmonic maps, meaning that half-harmonic maps give rise to a solution of the Plateau problem, but perhaps with multiplicity.

The second part of this thesis presents joint work with Melanie Rupflin and Michael Struwe which introduces a new system of equations. This flow is designed to produce free boundary minimal surfaces with topology of any fixed compact surface, with boundary supported on a smooth closed submanifold N of Rn. This is also a gradient flow for the half energy, but now in its generalised form as introduced by Da Lio and Pigati. We couple the equation for the map evolution with an equation to evolve the metric on the domain, which is inspired by the Teichmüller harmonic map flow introduced by Rupflin and Topping. For this system, we establish key results on existence and regularity of solutions, along with an analysis of singularity formation and asymptotic convergence.

In the final part of this thesis, I study a question about solutions of the Teichmüller harmonic map flow, which is a gradient flow of the Dirichlet energy with respect to both the map and domain metric which was introduced to produce closed minimal surfaces in a smooth closed target manifold (N, h). In particular, I study the fine structure of the limit object at infinite time in the setting where the domain metric degenerates and so part of the limiting map collapses to a curve. I study sufficient conditions to ensure that this curve is a geodesic in (N, h) for two slightly modified versions of the flow.

Authors: Wright, C

• DOI: 10.5287/ora-2z7bdpzy0
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: minimal surfaces; gradient flows; partial differential equations; half harmonic maps
• Subjects: Mathematics