Existence of Hyperbolic Calorons

Recent work of Harland shows that the $SO(3)$-symmetric, dimensionally-reduced, charge-$N$ self-dual Yang–Mills calorons on the hyperbolic space $H^3\times S^1$ may be obtained through constructing $N$-vortex solutions of an Abelian Higgs model as in the study of Witten on multiple instantons. In this paper we establish the existence of such minimal action charge-$N$ calorons by constructing arbitrarily prescribed $N$-vortex solutions of the Witten type equations.