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The existence of a class of cosmic string solutions with infinite energy

Published online by Cambridge University Press:  14 November 2011

Yisong Yang
Yisong Yang
Affiliation:
Department of Mathematics, University of New Mexico, Albuquerque, NM 87131, U.S.A.
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Synopsis

Recent studies have indicated that in a certain critical coupling phase the Einstein–matter–gauge equations may be reduced to a Bogomol'nyi–Einstein system and the solutions are cosmic strings with finite separation. In this paper, we view the spacetime as known and treat the dynamics as being determined by the matter–gauge sector in an asymptotically Euclidean geometry. It is shown that the model admits a continuous family of infinite energy solutions which decay to the symmetric vacuum faster than any exponential functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 301 - 315
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Aubin, T.. Nonlinear Analysis on Manifolds: Monge–Ampére Equations (New York: Springer, 1982).CrossRefGoogle Scholar
2Bogomol'nyi, E. B.. The stability of classical solutions. Soviet J. Nuclear Phys. 24 (1976), 449–454.Google Scholar
3Cantor, M.. Elliptic operators and the decomposition of tensor fields. Bull. Amer. Math. Soc. 5 (1981), 235–262.CrossRefGoogle Scholar
4Choquet-Bruhat, Y. and Christodoulou, D.. Elliptic systems in H sΔ spaces on manifolds which are Euclidean at infinity. Acta Math. 146 (1981), 129–150.CrossRefGoogle Scholar
5Comtet, A. and Gibbons, G. W.. Bogomol'nyi bounds for cosmic strings. Nuclear Phys. B 299 (1988), 719–733.CrossRefGoogle Scholar
6Comtet, A. and Houston, P. J.. Vortices and the prescribed curvature problem. J. Phys. A 18 (1985), L637–L641.CrossRefGoogle Scholar
7Gilbarg, D. and Trudinger, N.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1977).CrossRefGoogle Scholar
8Griffiths, P. and Harris, J.. Principles of Algebraic Geometry (New York: Wiley, 1978).Google Scholar
9Jaffe, A. and Taubes, C. H.. Vortices and Monopoles (Boston: Birkhäuser, 1980).Google Scholar
10Kibble, T., Lazarides, G. and Shaft, Q.. Strings in SO (10). Phys. Lett. B 113 (1982), 237–239.CrossRefGoogle Scholar
11Linet, B.. A vortex-line model for infinite straight cosmic strings. Phys. Lett. A 124 (1987), 240–242.CrossRefGoogle Scholar
12Linet, B.. A vortex-line model for a system of cosmic strings in equilibrium. General Relativity Gravitation 20 (1988), 451–456.CrossRefGoogle Scholar
13Linet, B.. On the supermassive U (1) gauge cosmic strings. Classical Quantum Gravity 7 (1990), L75–L79.CrossRefGoogle Scholar
14Lockhart, R.. Fredholm properties of a class of elliptic operators on non-compact manifolds. Duke Math. J. 48 (1981), 289–312.CrossRefGoogle Scholar
15McOwen, R.. The behavior of the Laplacian on weighted Sobolev spaces. Comm. Pure Appl. Math. 32 (1979), 783–795.CrossRefGoogle Scholar
16McOwen, R.. On the equation Δu+Ke 2u = f and prescribed negative curvature in ℝ2. J. Math. Anal. Appl. 103 (1984), 365–370.CrossRefGoogle Scholar
17McOwen, R.. Conformal metric in ℝ2 with prescribed Gaussian curvature and positive total curvature. Indiana Univ. Math. J. 34 (1985), 97–104.CrossRefGoogle Scholar
18Ni, W.-M.. On the elliptic equation Δu+K(x)e 2u = 0 and conformal metrics with prescribed Gaussian curvatures. Invent. Math. 66 (1982), 343–352.CrossRefGoogle Scholar
19Parker, T. and Taubes, C. H.. On Witten's proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 84 (1982), 223–238.CrossRefGoogle Scholar
20Schoen, R. and Yau, S. T.. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65 (1979), 45–76.CrossRefGoogle Scholar
21Vilenkin, A.. Cosmic strings and domain walls. Phys. Rep. 121 (1985), 263–315.CrossRefGoogle Scholar
22Wang, S. and Yang, Y.. Abrikosov's vortices in the critical coupling. SIAM J. Math. Anal. (to appear).Google Scholar
23Yang, Y.. Vortices on asymptotically Euclidean Riemann surfaces. Nonlinear Anal. 15 (1990), 577–596.CrossRefGoogle Scholar
24Zeldovich, Ya. B.. Mon. Not. R. Astron. Soc. 192 (1980), 663.CrossRefGoogle Scholar