Affine Harmonic Maps

TL;DR

This paper introduces affine harmonic maps from affine flat manifolds to Riemannian manifolds, establishing existence results under curvature conditions despite the lack of a variational structure.

Contribution

It defines a new class of maps called affine harmonic maps and proves their existence in certain homotopy classes with non-positive curvature targets.

Findings

Existence of affine harmonic maps under specific curvature conditions

Necessity of non triviality condition demonstrated by example

Development of estimation techniques for non-variational elliptic systems

Abstract

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic maps in a given homotopy class when the target has non positive sectional curvature and some global non triviality condition is met. An example shows that such a condition is necessary. The analytical part is made difficult by the absence of a variational structure underlying affine harmonic maps. We therefore need to combine estimation techniques from geometric analysis and PDE theory with global geometric considerations.