Polarized harmonic mappings and optimal moving frames

Abstract

We introduce the notion of a polarized harmonic mapping M → N of Riemannian manifolds and its infinitesimal analogue, a polarized harmonic connection. We study the integrability of these polarized harmonic connections when M is foliated by the action of a Lie group G.  In the case that M is a Kähler manifold and G has dimension 1, the polarized harmonic connection is integrable and we obtain a polarized harmonic mapping M → G, known as an optimal moving frame (or optimal G-frame). These ideas are illustrated using the dynamical system of N-point vortices in the plane.