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RICCI FLOWS ON SURFACES RELATED TO THE EINSTEIN WEYL AND ABELIAN VORTEX EQUATIONS

Published online by Cambridge University Press:  22 August 2014

DANIEL J. F. FOX
DANIEL J. F. FOX*
Affiliation:
Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012 Madrid España e-mail: [email protected]
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Abstract

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There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 3 , September 2014 , pp. 569 - 599
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Bakas, I., Renormalization group flows and continual Lie algebras, J. High Energy Phys. (8) (2003), 013, 70 pp.Google Scholar
2.Bernstein, J. and Mettler, T., Two-dimensional gradient Ricci solitons revisited, arXiv:1303.6854 (2013).CrossRefGoogle Scholar
3.Bradlow, S. B., Vortices in holomorphic line bundles over closed Kähler manifolds, Comm. Math. Phys. 135 (1) (1990), 1–17.Google Scholar
4.Calderbank, D. M. J., Möbius structures and two-dimensional Einstein-Weyl geometry, J. Reine Angew. Math. 504 (1998), 37–53.CrossRefGoogle Scholar
5.Calderbank, D. M. J., Two dimensional Einstein-Weyl structures, Glasg. Math. J. 43 (3) (2001), 419–424.Google Scholar
6.Chen, B.-L. and Zhu, X.-P., Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differ. Geom. 74 (1) (2006), 119–154.Google Scholar
7.Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F. and Ni, L., The Ricci flow: techniques and applications. Part II, in Mathematical Surveys and Monographs, vol. 144 (American Mathematical Society, Providence, RI, 2008), Analytic aspects.Google Scholar
8.Daskalopoulos, P., Hamilton, R. and Sesum, N., Classification of ancient compact solutions to the Ricci flow on surfaces, J. Differ. Geom. 91 (2) (2012), 171–214.Google Scholar
9.Daskalopoulos, P. and Sesum, N., Eternal solutions to the Ricci flow on ℝ2, Int. Math. Res. Not. (2006), Art. ID 83610, 20.Google Scholar
10.Farkas, H. M. and Kra, I., Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71 (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar
11.Fateev, V. A., Onofri, E. and Zamolodchikov, A. B., Integrable deformations of the O(3) sigma model. The sausage model, Nucl. Phys. B 406 (3) (1993), 521–565.Google Scholar
12.Fox, D. J. F., Einstein-like geometric structures on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. XII (5) (2013), 499–585.Google Scholar
13.García-Prada, O., Invariant connections and vortices, Comm. Math. Phys. 156 (3) (1993), 527–546.Google Scholar
14.García-Prada, O., A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1) (1994), 88–96.Google Scholar
15.Hamilton, R. S., The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71, Amer. Math. Soc. (Providence, RI, 1988), 237–262.Google Scholar
16.Hamilton, R. S., The formation of singularities in the Ricci flow, in Surveys in differential geometry, vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, 7–136.Google Scholar
17.Jaffe, A. and Taubes, C., Vortices and monopoles, in Progress in physics, vol. 2 (Birkhäuser Boston, Mass., 1980), Structure of static gauge theories.Google Scholar
18.Kazdan, J. L. and Warner, F. W., Curvature functions for compact 2-manifolds, Ann. Math. 99 (2) (1974), 14–47.CrossRefGoogle Scholar
19.King, J. R., Exact polynomial solutions to some nonlinear diffusion equations, Phys. D 64 (1–3) (1993), 35–65.Google Scholar
20.Manton, N. S., One-vortex moduli space and Ricci flow, J. Geom. Phys. 58 (12) (2008), 1772–1783.Google Scholar
21.Manton, N. S., Vortex solutions of the Popov equations, J. Phys. A: Math. Theor. 46 (14) (2013), 145402.Google Scholar
22.Noguchi, M., Yang-Mills-Higgs theory on a compact Riemann surface, J. Math. Phys. 28 (10) (1987), 2343–2346.Google Scholar
23.Popov, A. D., Integrable vortex-type equations on the two-sphere, Phys. Rev. D 86 (2012), 105044.Google Scholar
24.Ramos, D., Gradient Ricci solitons on surfaces, arXiv:1304.6391 (2013).Google Scholar
25.Rosenau, P., Fast and superfast diffusion processes, Phys. Rev. Lett. 74 (7) (1995), 1056–1059.Google Scholar
26.Tashiro, Y., Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275.Google Scholar
27.Taubes, C. H., On the equivalence of the first and second order equations for gauge theories, Comm. Math. Phys. 75 (3) (1980), 207–227.Google Scholar
28.Uhlenbeck, K. K., Closed minimal surfaces in hyperbolic 3-manifolds, in Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103 (Princeton Univ. Press, Princeton, NJ, 1983), 147–168.Google Scholar
29.Witten, E., Some exact multipseudoparticle solutions of classical Yang-Mills theory, Phys. Rev. Lett. 38 (3) (1977), 121–124.Google Scholar
30.Wu, L.-F., The Ricci flow on 2-orbifolds with positive curvature, J. Differ. Geom. 33 (2) (1991), 575–596.Google Scholar