Non-divergence harmonic maps

TL;DR

This paper investigates non-divergence elliptic and parabolic systems on manifolds, including Hermitian and affine harmonics, emphasizing the need for topological conditions for solution existence beyond curvature constraints.

Contribution

It introduces a framework for non-variational harmonic maps on manifolds, highlighting the role of topological conditions alongside curvature assumptions.

Findings

Solutions depend on global topological conditions.

Includes analysis of Hermitian and affine harmonic systems.

Extends harmonic map theory beyond variational settings.

Abstract

We describe work on solutions of certain non-divergence type and therefore non-variational elliptic and parabolic systems on manifolds. These systems include Hermitian and affine harmonics which should become useful tools for studying Hermitian and affine manifolds, resp. A key point is that in addition to the standard condition of nonpositive image curvature that is well known and understood in the theory of ordinary harmonic maps (which arise from a variational problem), here we also need in addition a global topological condition to guarantee the existence of solutions.