Title:

Geometric flows on soliton moduli spaces

It is well known that the low energy dynamics of many types of soliton can be approximated by geodesic motion on Mn, the moduli space of static nsolitons, which is usually a Kähler manifold. This thesis presents a detailed study of magnetic geodesic motion on a Kähler manifold in the case where the magnetic field 2form is the Ricci form. This flow, which we call Ricci Magnetic Geodesic (RMG) flow, is first studied in general. A symmetry reduction result is proved which allows one to localize the flow onto the fixed point set of any group of holomorphic isometries of a Kähler manifold M. A subtlety of this reduction, which was overlooked by Krusch and Speight, is pointed out. Since RMG flow occurs at constant speed, it follows immediately that the flow is complete if M is geodesically complete. We show, by means of an explicit counterexample that, contrary to a conjecture of Krusch and Speight, the converse is false: it is possible for a geodesically incomplete manifold to be RMG complete. RMG completeness of metrically incomplete manifolds is therefore a nontrivial issue, and one which will be addressed repeatedly in later chapters. We then specialize to the case where Mn is the moduli space of abelian Higgs nvortices, which is the context in which RMG flow was first proposed, by Collie and Tong, as a low energy model of the dynamics of a certain type of ChernSimons nvortices on ℝ2. The unit vortex is constructed numerically, and its asymptotics is studied. It is shown that, contrary to an assertion of Collie and Tong, RMG flow does not coincide with an earlier proposed magnetic geodesic model of vortex motion due to Kim and Lee. It is further shown that Kim and Lee’s model is illdefined on the vortex coincidence set. An asymptotic formula for the scattering angle of wellseparated vortices executing RMG flow is computed. We then change the spatial geometry, placing the vortices on the hyperbolic plane of critical curvature. An explicit formula for the twovortex metric is derived, extending the results of Strachan, who computed the metric on a submanifold of centred 2vortices. The RMG flow localized on this submanifold is compared with its intrinsic RMG flow, revealing strong qualitative differences. We then study the moduli space Hn,k(∑) of degree n ℂPk lumps on a compact Riemann surface ∑. It is shown that Rat1 = H1,1(S2) is RMG complete (despite being metrically incomplete). The EinsteinHilbert action of H1,k(S2) is computed, supporting (for k > 1) a conjecture of Baptista. A natural class of topologically cylindrical submanifolds of Hn,1(∑), called dilation cylinders, is studied: their volumes are computed, and it is shown that they are all isometrically embeddable as surfaces of revolution in R3. Conditions under which they are totally geodesic, for ∑ = S2 and T2, are found, and RMG flow on some examples is studied. Finally, a new metric on Hn,1(∑), derived from the BabySkyrme model, is introduced. On Rat1, this metric is determined explicitly and some geometric aspects such as the volume, geodesic flow and the spectral problem with respect to this metric are studied.
