Mathematical Research Letters

Volume 5 (1998)

Number 4

Character varieties and harmonic maps to ${\Bbb R}$-trees

Pages: 523 – 533

DOI: http://dx.doi.org/10.4310/MRL.1998.v5.n4.a9

Authors

G. Daskalopoulos (Brown University)

S. Dostoglou (University of Missouri)

R. Wentworth (University of California at Irvine)

Abstract

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\Bbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\Bbb R}$-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.

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