Analysis of boundary bubbles for almost minimal cylinders

TL;DR

This paper investigates the asymptotic behavior of solutions to the Teichmüller harmonic map flow from cylinders under degenerating boundary conditions, showing that boundary bubbles form as branched minimal immersions, completing the understanding of flow convergence.

Contribution

It proves that boundary bubbles in degenerating boundary conditions are harmonic branched minimal immersions, advancing the analysis of the Teichmüller harmonic map flow for almost minimal cylinders.

Findings

Boundary bubbles form as the domain stretches during degenerating boundary conditions.

Boundary bubbles are harmonic branched minimal immersions under Douglas' separation condition.

The work completes the proof that the flow transforms initial surfaces into solutions of the Douglas-Plateau problem.

Abstract

We analyse the asymptotic behaviour of solutions of the Teichm\"uller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition forces the domain to stretch out as a boundary bubble forms. Our main result then establishes that for prescribed boundary curves that satisfy Douglas' separation condition, these boundary bubbles will not only be harmonic but will themselves be branched minimal immersions. Together with earlier work, this in particular completes the proof that the Teichm\"uller harmonic map flow changes every initial surface in $\mathbb{R}^n$ spanning such boundary curves into a solution of the corresponding Douglas-Plateau problem.