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**Extra resources for Curves and Abelian Varieties: International Conference March 30-april 2, 2007 University of Georgia Athens, Georgia**

**Sample text**

If ~ < fl < 1, we may also write E(vls,/) = E(vls~;) + E(Vls,~\s;). Since the energy of ue on S'~ \ S'~ again is bounded from below by the corresponding one of v up to some additive constant c4, we obtain from the energy minimizing property of u~ E(~I~,) _< E(v~ls~,) + ~, with c5 independent of 5 (but not of r]). Thus, the energy of u~ on S~ for fixed 7] > 5 is bounded independently of ~. Therefore, as d --~ 0, some subsequence of u~ converges on S~' to a harmonic map, using standard estimates for harmonic maps (see [H3]).

2 Let Y be a complete, simply connected, locally compact space of nonpositive curvature with isometry group I(Y). Let F be a group, p : P -+ I(Y) a homomorphism. e. is not contained in a parabolic subgroup of the isometry group of Z'. In geometric terms, this means t h a t there does not exist an unbounded sequence (Yn)neN C Z' with dist(yn, 7(Yn)) --< C(7) for all 7 E p(F), with a constant depending on y, but not on n. Note that any semisimple representation is reductive. For simplicity of notation, we shall consider u in the sequel as a m a p from X into N := y / p ( ~ l ( x ) ) , although N may be singular.

Since u has finite energy, almost all such restrictions need to have finite energy, and as in [JY2, p. 302] we then see that in fact all of them do. Case 1. Suppose that z0 C D ~ lies in the smooth part of Do~. Therefore, estimates for u follow from estimates for finite energy harmonic maps h: D*-+ N where D* -- {z C C : 0 < [z I < 1} is the punctured uinit disk, and N is a R i e m mannian manifold of nonpositive sectional curvature. In the sequel, D* will always be equipped with the Euclidean metric of C, and (r, 9~) will be standard polar coordinates.