a b s t r a c tAnti-self-dual (ASD) solutions to the Yang-Mills equation (or instantons) over an antiself-dual 4-manifold, which are invariant under an appropriate action of a threedimensional Lie group, give rise, via twistor construction, to isomonodromic deformations of connections on CP 1 having four simple singularities. As is well known, such deformations are governed by the sixth Painlevé equation Pvi (α, β, γ , δ). We work out the particular case of the SU 2 -action on S 4 , obtained from the irreducible representation on R 5 . In particular, we express the parameters (α, β, γ , δ) in terms of the instanton number. The present paper contains the proof of the result announced in [Richard Muñiz Manasliski, Painlevé VI equation from invariant instantons, in: Geometric and Topological Methods for
We give a new proof of the uniformization theorem of the leaves of a compact lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversally continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.
Abstract. We consider a two parameter family of instantons, which is studied in [Sadun L., Comm. Math. Phys. 163 (1994), 257-291], invariant under the irreducible action of SU 2 on S 4 , but which are not globally defined. We will see that these instantons produce solutions to a one parameter family of Painlevé VI equations (P VI ) and we will give an explicit expression of the map between instantons and solutions to P VI . The solutions are algebraic only for that values of the parameters which correspond to the instantons that can be extended to all of S 4 . This work is a generalization of [Muñiz Manasliski R
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