Energy of Twisted Harmonic Maps of Riemann Surfaces

Abstract

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_\rho$ on Teichm\"uller space $\Teich$ which is a qualitative invariant of the holonomy representation $\rho$ of $\pi_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function $E_\rho$ is proper for any convex cocompact representation of the fundamental group. More generally, if $\rho$ is a discrete embedding onto a normal subgroup of a convex cocompact group $\Gamma$, then $E_\rho$ defines a proper function on the quotient $\Teich/Q$ where $Q$ is the subgroup of the mapping class group defined by $\Gamma/\rho(\pi_1(S))$. When the image of $\rho$ contains parabolic elements, then $E_\rho$ is not proper. Using the recent solution of Marden's Tameness Conjecture, we show that if $\rho$ is a discrete embedding into $\SLtC$, then $E_\rho$ is proper if and only if $\rho$ is quasi-Fuchsian. These results are used to prove that the mapping class group acts properly on the subset of convex cocompact representations.