Limiting behaviour of the Teichmuller harmonic map flow

• Main Author: Huxol, Tobias
• Published: University of Warwick 2016
• Subjects: 515; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.694629

In this thesis we study two problems related to the Teichm�uller harmonic map flow, a flow introduced in [21], which aims to deform maps from closed surfaces into closed Riemannian targets of general dimension into branched minimal immersions. It arises as a gradient flow for the energy functional when one varies both the map and the domain metric. We first consider weak solutions of the flow that exist for all time, with metric degenerating at in�nite time. It was shown in [25] that for such solutions one can extract a so-called sequence of almost-minimal maps, which subconverges to a collection of branched minimal immersions (or constant maps). We further improve this compactness theory, in particular showing that no loss of energy can happen, after accounting for all developing bubbles. We also construct an example of a smooth flow where the image of the limit branched minimal immersions is disconnected. These results were obtained in [13], joint with Melanie Ruping and Peter Topping. Secondly, we study limits of the coupling constant n, which controls the relative speed of the metric evolution and the map evolution along the flow. We show that when n | 0, corresponding to slowing down the metric evolution, one obtains the classical harmonic map flow as a limit of the Teichm�uller harmonic map flow when the target N has nonpositive sectional curvature. Finally, we let N | 0 and simultaneously rescale time, 'fixing' the speed at which the metric evolves and accelerating the evolution of the map component. We show that in this setting the Teichm�uller harmonic map flow converges to a flow through harmonic maps, if one assumes the target N to have strictly negative sectional curvature everywhere and the initial map to not be homotopic to a constant map or a map to a closed geodesic in the target.