Proper Actions of the Mapping Class Group and the Energy of Harmonic Maps

W. Goldman and R. Wentworth

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over Riemann surfaces is a function on Teichmueller space which is a qualitative invariant of the holonomy representation. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function is proper for any convex cocompact representation of the fundamental group. As a corollary, the mapping class group acts properly on the subset of convex cocompact representations.