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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.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32J27 Compact Kähler manifolds: generalizations, classification
32Q20 Kähler-Einstein manifolds
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