The harmonic map heat flow from surfaces
We present a study of the harmonic map heat flow of Eells and Sampson in the case that the domain manifold is a surface. Particular emphasis has been placed on the singularities which may occur, as described by Struwe, and the analysis of the flow despite these. In Chapter 1 we give a brief introduction to the theory of harmonic maps and their flow. Further details are to be found in  and . In the case that the domain manifold is a surface we describe the existence theory for the heat flow and the theory of bubbling. In Chapter 2 we investigate the question of the uniformity in time of the convergence of the heat flow to a bubble tree at infinite time. In Section (2.1) (page 28) we give the first example of a non-uniform flow. In contrast, Theorem (2.2) (page 30) provides conditions under which the convergence is uniform and any bubbles which form are rigid. In Chapter 3 we give the first example of a nontrivial bubble tree - in other words we give a flow in which more than one bubble develops at the same point at infinite time. In Chapter 4 we discuss in what sense two flows are close when their initial maps are close. We formulate this question in various ways, providing examples of instability and an `infinite time' stability result (Theorem (4.2), page 56) using techniques developed in Chapter 2. From the theory of bubbling as described in Chapter 1, if an initial map has less energy than is required for a bubble, then the subsequent flow cannot blow up. In Chapter 5 we ask conversely whether given enough energy for a bubble, we can find an initial map leading to blow-up. In the appendix we outline a plausible construction of a flow which can be analysed at two different sequences of times to give convergence to two different bubble trees, with different numbers of bubbles.