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Geometric flows on soliton moduli spaces

Alqahtani, Lamia Saeed M (2013) Geometric flows on soliton moduli spaces. PhD thesis, University of Leeds.

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Abstract

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 n-solitons, 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 2-form 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 n-vortices, 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 Chern-Simons n-vortices 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 ill-defined on the vortex coincidence set. An asymptotic formula for the scattering angle of well-separated 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 two-vortex metric is derived, extending the results of Strachan, who computed the metric on a submanifold of centred 2-vortices. 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 Einstein-Hilbert 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 Baby-Skyrme 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.

Item Type: Thesis (PhD)
ISBN: 978-0-85731-509-0
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.589303
Depositing User: Repository Administrator
Date Deposited: 09 Jan 2014 13:29
Last Modified: 07 Mar 2014 11:48
URI: http://etheses.whiterose.ac.uk/id/eprint/4967

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