"Affine harmonic maps"

By

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**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 Riemannian 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 nonpositive sectional curvature and
some global nontriviality 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.

Tea and cookies will be served before the
seminar.

**Date: ****Thursday,
April 9, 2009**

**Time: ****14.40**

**Place: ****Mathematics Seminar Room, SA-141**