Global weak solutions of the Teichmüller harmonic map flow into general targets

Rupflin, Melanie; Topping, Peter M.

doi:10.2140/apde.2019.12.815

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Global weak solutions of the Teichmüller harmonic map flow into general targets

Melanie Rupflin and Peter M. Topping

Vol. 12 (2019), No. 3, 815–842

DOI: 10.2140/apde.2019.12.815

Abstract

We analyse finite-time singularities of the Teichmüller harmonic map flow — a natural gradient flow of the harmonic map energy — and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves.

Keywords

geometric flows, minimal surfaces, harmonic maps

Mathematical Subject Classification 2010

Primary: 53A10, 53C43, 53C44

Milestones

Received: 19 December 2017

Accepted: 29 June 2018

Published: 7 October 2018

Authors

Melanie Rupflin
Mathematical Institute University of Oxford Oxford United Kingdom

Peter M. Topping
Mathematics Institute University of Warwick Coventry United Kingdom

Vol. 12, No. 3, 2019