Lojasiewicz inequalities for the harmonic map energy

In this thesis we study the Lojasiewicz-Simon inequality, a fundamental tool in studying the asymptotic behaviour of gradient flows and of almost-critical points. We focus on the harmonic map energy of maps from surfaces into closed Riemannian target manifolds, viewed as a functional of both map and a domain metric. In the first part of this thesis we establish new results for the Teichmuller harmonic map flow, an L^2 gradient flow for the harmonic map energy that was introduced by Rupflin and Topping. Previous theory established that in the absence of singularities the flow converges to a branched minimal immersion, but only after passing to a subsequence of times going to infinity and pulling back by a sequence of diffeomorphisms. We show that in general both passing to a sub-sequence and pulling back by diffeomorphisms are actually necessary to obtain convergence of the flow by constructing several “winding” solutions of the flow from the torus. Next, when the target manifold is real-analytic we establish a Lojasiewicz-Simon inequality and use it to show that the flow converges as time goes to infinity, that is without the need to pass to a subsequence or pull back by diffeomorphisms, provided it does not encounter singularities. These results are joint work with Rupflin and Topping. In the second part of this thesis we study the harmonic map energy of maps from degenerating hyperbolic cylinders into general closed Riemannian target manifolds, motivated by the singular behaviour that can occur for the metric component of the Teichmuller harmonic map flow. We prove a Lojasiewicz inequality with optimal exponent for sequences of almost-critical points which have fixed boundary values and which do not form bubbles, provided their limit satisfies a non-degeneracy condition on the second variation of the energy. Such sequences undergo a change of topology in the limit meaning that the powerful techniques introduced by Simon do not apply. We instead use an approach introduced by Malchiodi, Rupflin and Sharp and obtain a Lojasiewicz inequality in the neighbourhood of a carefully constructed family of “adapted critical points”.