Character varieties and harmonic maps to ℝ-trees

Abstract

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible SL2 (ℂ) representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ℝ-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.