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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References:

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