Affine Harmonic Maps
J\"urgen Jost (1), Fatma Muazzez \c{S}im\c{s}ir (2) ((1) Max Planck, Institute for the Mathematics in the Sciences, (2) TOBB University of, Economics, Technology)

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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Numerical methods in inverse problems · Nonlinear Partial Differential Equations
