Energy of Twisted Harmonic Maps of Riemann Surfaces

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface S is a function Eρ on Teichmüller space TS which is a qualitative invariant of the holonomy representation ρ of π1(S). Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function Eρ is proper for any convex cocompact representation of the fundamental group. More generally, if ρ is a discrete embedding onto a normal subgroup of a convex cocompact group Γ, then Eρ defines a proper function on the quotient TS/Q where Q is the subgroup of the mapping class group defined by Γ/ρ(π1(S)). When the image of ρ contains parabolic elements, then Eρ is not proper. Using the recent solution of Marden’s Tameness Conjecture, we show that if ρ is a discrete embedding into SL(2,C), then Eρ is proper if and only if ρ is quasi-Fuchsian. These results are used to prove that the mapping class group acts properly on the subset of convex cocompact representations.