Geometry of solutions of Hitchin equations on R^2.

Abstract

We study smooth SU(2) solutions of the Hitchin equations on ${{\mathbb{R}}^{2}}$ , with the determinant of the complex Higgs field being a polynomial of degree n. When $n\geqslant 3$ , there are moduli spaces of solutions, in the sense that the natural L 2 metric is well-defined on a subset of the parameter space. We examine rotationally-symmetric solutions for n  =  1 and n  =  2, and then focus on the n  =  3 case, elucidating the moduli and describing the asymptotic geometry as well as the geometry of two totally-geodesic surfaces.