Title:

A study of the heat flow for closed curves with applications to geodesics

A description of the material presented in this thesis. Throughout it is concerned with closed curves i.e. the domain M above is replaced by the unit circle S1. Part I is concerned with the problem of carrying out the method described above in the case of domain S1, without imposing any curvature restrictions on the target manifold M, and without any conditions depending on an embedding of the manifold. This problem is approached from an angle slightly different from that of Eells and Sampson as follows: first, for a given closed C1 curve f, a condition on M will be given that will ensure that any solution fs of the heat equation, which is continuous along with its first S1  derivative and which coincides with f at s = 0 has its image contained in a fixed compact subset of M. This condition is different from the one in [ES] and does not depend on an embedding. Then, assuming that that is the case (i.e. all solutions uniformly bounded) the existence of a unique solution defined for all positive values of the deformation parameter will be proved. The proof of this result will follow the corresponding proof in [ES] but the conditions on M (in particular the curvature restriction) used there will not be necessary. Finally, in Part I, it is proved that this solution subconverges to a closed geodesic, again following [ES] but without the curvature restriction. The assertion that the results of [ES] hold for closed curves on compact Riemannian manifolds without curvature restrictions was made in "Variational theory in fibre bundles" by J. Eells and J.H. Sampson, Proc. USJapan Sem. Diff. Geo. Kyoto (1965) 2223 and in "On harmonic mappings" by J.H. Sampson, Istituto Nazionale di Alta Matematica Francesco Severi Symposia Matematica Vol. XXVI (1982). The latter contains some remarks about the proof. The proof in this case (i.e. for closed curves on compact Riemannian manifolds of arbitrary curvature) formed the author's M.Sc. dissertation written at Warwick University in the academic year 198182. Part I appeared in the Journal of Geometry and Physics Vol. 2, n.1, 1985 under the title "Closed geodesics on Riemannian manifolds via the heat flow". Part II is taken up with a discussion about the question : What happens if one considers Lorentz manifolds instead of Riemannian ones? More specifically, given a closed curve f on a Lorentz manifold M, can one prove results about existence, convergence etc. of a solution fs of the heat equation with fo = f, of the same kind as those in [ES]? The change from a positive definite metric to a nondefinite one turned out to change the properties of the solutions of the heat equation as is illustrated by some examples in Section 5 of Part II. For instance, there are examples of solutions for which the energy is not bounded and examples of solutions which only exist up to a finite value of the deformation parameter. However, there are also examples of solutions which exist for all s ˃ 0, are bounded along with their derivatives and which converge uniformly to a closed geodesic. Part II is then devoted to the investigation of to what extent one can carry out the method from [ES] described above. In more detail as follows: The idea that one could apply the results already obtained in Part I lead, to the study of certain Riemannian metrics naturally associated to timeorientable Lorentz manifolds. However, the result was that this approach only works in the special case where the Lorentz manifold has a parallel timelike vector field. As to the causal character of curves, it turned out that applying heat flow to closed timelike curves, the solution will preserve that property, i.e. will define a thomotopy of the initial curve. The property of being a spacelike or lightlike closed curve is, however, in general, not preserved. The simple property of solutions on Riemannian manifolds that the energy of the curves fs decreases as s increases does not have an analogue on Lorentz manifolds (see examples III and IV of Section 5, Part II). For timelike solutions however, one can say a bit more both about the evolution of the energy and of the "length" as defined by the Lorentz metric, in fact, the length of the curves fs for such solutions increases as s increases. As for the existence of solutions, the main difficulty lies in the fact that one cannot use the energy density, in the same way as in [ES] and Part I, as a measure on the first derivative of the curves fs, both because the energy density is, in general, not bounded, and even if it were, that would, because the metric is nondefinite, not imply boundedness in any Riemannian metric. The proofs of uniqueness and of existence for small values of the deformation parameter are the same as for Riemannian metrics, but in order to prove existence for all positive values it was necessary to make certain new boundedness assumptions on the first θderivative of the solution. Finally, in Part II there is a proof of subconvergence of bounded timelike solutions with bounded first 9derivative to a closed geodesic. This thesis has benefitted from the valuable advice and guidance of Professor J. Eells.
