On finite energy monopoles on ℂ× Σ

2018/11/07 by Wang, Donghao
#57M50 #57R57 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1811.03139

Abstract

Let X=ℂ×Σ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on X with finite analytic energy. The spin bundle S⁺→ X splits as L⁺⊕ L^-. When 2-2g≤ c₁(S⁺)[Σ]<0, the moduli space is in bijection with the moduli space of pairs ((L⁺,∂), f) where (L⁺,∂) is a holomorphic structure on L⁺ and f: ℂ→ H⁰(Σ, L⁺,∂) is a polynomial map. Moreover, the solution has analytic energy -4π²d⋅ c₁(S⁺)[Σ] if f has degree d. When c₁(S⁺)=0, all solutions are reducible and the moduli space is the space of flat connections on \bigwedge² S⁺. We also estimate the decay rate of these solutions at infinity.

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