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SEIBERG–WITTEN FLOW IN HIGHER DIMENSIONS

Part of: Partial differential equations

Published online by Cambridge University Press:  01 March 2013

LORENZ SCHABRUN
LORENZ SCHABRUN*
Affiliation:
Department of Mathematics, The University of Queensland, Brisbane, Qld 4072, Australia (email: lorenz@maths.uq.edu.au)
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Abstract

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We show that for manifolds of dimension $m\geq 5$, the flow of a Seiberg–Witten-type functional admits a global smooth solution.

Keywords

Seiberg–Witten flowglobal existencehigher dimensions

MSC classification

Secondary: 35A01: Existence problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 3 , December 2012 , pp. 311 - 324
Copyright
Copyright © 2013 Australian Mathematical Publishing Association Inc. 

References

[1]Bilge, A. H., Dereli, T. and Kocak, S., ‘Seiberg–Witten type monopole equations on 8-manifolds with spin(7) holonomy, as minimizers of a quadratic action’, J. High Energy Phys. 4(003).Google Scholar
[2]Değirmenci, N. and Özdemir, N., ‘Seiberg–Witten-like equations on 7-manifolds with $g_ 2$-structure’, J. Nonlinear Math. Phys. 12(4) (2005), 457–461.CrossRefGoogle Scholar
[3]Evans, L. C., Partial Differential Equations (American Mathematical Society, Providence, RI, 1998).Google Scholar
[4]Gao, Y.-H. and Tian, G., ‘Instantons and the monopole-like equations in eight dimensions’, J. High Energy Phys. 5(36).Google Scholar
[5]Hong, M.-C. and Schabrun, L., ‘Global existence for the Seiberg–Witten flow’, Comm. Anal. Geom. 18(3) (2010), 433–474.CrossRefGoogle Scholar
[6]Hong, M.-C. and Tian, G., ‘Asymptotical behaviour of the Yang–Mills flow and singular Yang–Mills connections’, Math. Ann. 330 (2004), 441–472.CrossRefGoogle Scholar
[7]Jost, J., Riemannian Geometry and Geometric Analysis (Springer, Berlin, 1995).CrossRefGoogle Scholar
[8]Lin, F. H., ‘Gradient estimates and blow-up analysis for stationary harmonic maps’, Ann. of Math. 149 (1999), 785–829.CrossRefGoogle Scholar
[9]Lin, F. H. and Wang, C. Y., ‘Harmonic and quasi-harmonic spheres’, Comm. Anal. Geom. 10(2) (1999), 397–429.CrossRefGoogle Scholar
[10]Peng, X., Jost, J. and Wang, G., ‘Variational aspects of the Seiberg–Witten functional’, Calc. Var. Partial Differential Equations 3 (1996), 205–218.Google Scholar
[11]Scorpan, A., The Wild World of 4-Manifolds (American Mathematical Society, Providence, RI, 2005).Google Scholar
[12]Shen, C.-L. and Chen, Y., ‘Monotonicity formula and small action regularity for Yang–Mills flows in higher dimensions’, Calc. Var. Partial Differential Equations 2(4) (1994), 389–403.Google Scholar
[13]Struwe, M., ‘On the evolution of harmonic maps in higher dimensions’, J. Diff. Geom. 28 (1999), 485–502.Google Scholar
[14]Tian, G., ‘Gauge theory and calibrated geometry, I’, Ann. of Math. 151 (2000), 193–268.CrossRefGoogle Scholar
[15]Uhlenbeck, K., ‘Connections with $l^p$-bounds on curvature’, Commun. Math. Phys. 83 (1982), 31–42.CrossRefGoogle Scholar