Seiberg Witten invariants and uniformizations. (English) Zbl 0856.53034

We study the Seiberg Witten equations and their applications in uniformization problems. First, we show that Kähler surfaces covered by a product of disks can be characterized using negative Seiberg Witten invariant. Second, we use Seiberg Witten equations to construct projectively flat \(U(2,1)\) connections on Einstein manifolds and uniformize those with optimal Chern numbers. Third, we study \(\Omega^2_-\) invariant solutions for \(U(2,1)\) perturbed anti-self-dual equations and show that they decouple to the Seiberg Witten equations and Einstein equations.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32J27 Compact Kähler manifolds: generalizations, classification
32Q20 Kähler-Einstein manifolds
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[1] [AHS] M. Atiyah, N. Hitchin, I. Singer: Self duality in four dimensional Riemannian geometry. Proc. Roy. Soc. London Ser.A362 (1978), 425-461 · Zbl 0389.53011
[2] [H] N. Hitchin: The self duality equations on a Riemann surface. Proc. London Math. Soc. 355 (1987), 59-126 · Zbl 0634.53045
[3] [J+Y] J. Jost, S.T. Yau: A strong rigidity theorem for a certain class of compact complex analytic surfaces. Math. Ann.271 (1985), 143-152 · Zbl 0554.32021
[4] [K+M] P. Kronheimer, T. Mrowka: The genus of embedded surfaces in the complex projective plane. Math. Res. Lett. · Zbl 0851.57023
[5] [L+M] B. Lawson, M. Michelsohn: Spin Geometry. Princeton University Press, (1989)
[6] [L] C. LeBrun: Einstein metrics and Mostow Rigidity. Math. Res. Lett. · Zbl 0974.53035
[7] [S] C. Simpson: Construction variations of Hodge structure using Yang-Mills theory, with applications to uniformation. J. Amer. Math. Soc.1 (1989), 867-918 · Zbl 0669.58008
[8] [W] E. Witten: Monopoles and four-manifolds. · Zbl 0867.57029
[9] [Y] S.T. Yau: On the Ricci curvature of a compact Kähler manifold and the complex Mönge-Ampère equation. I. C.P.A.M.31 (1978), 339-411 · Zbl 0369.53059
[10] [Y2] S.T. Yau: Splitting theorem and an algebraic geometric characterization of locally Hermitian symmetric spaces. Comm. Anal. Geom.1 (3) (1993), 473-486 · Zbl 0842.53035
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