The energy spectrum of metrics on surfaces

Abstract

Let $(N,\rho)$ be a Riemannian manifold, $S$ a surface of genus at least two and let $f\colon S \to N$ be a continuous map. We consider the energy spectrum of $(N,\rho)$ (and $f$) which assigns to each point $[J]\in \mathcal{T}(S)$ in the Teichm\"uller space of $S$ the infimum of the Dirichlet energies of all maps $(S,J)\to (N,\rho)$ homotopic to $f$. We study the relation between the energy spectrum and the simple length spectrum. Our main result is that if $N=S$, $f=id$ and $\rho$ is a metric of non-positive curvature, then the energy spectrum determines the simple length spectrum. Furthermore, we prove that the converse does not hold by exhibiting two metrics on $S$ with equal simple length spectrum but different energy spectrum. As corollaries to our results we obtain that the set of hyperbolic metrics and the set of singular flat metrics induced by quadratic differentials satisfy energy spectrum rigidity, i.e. a metric in these sets is determined, up to isotopy, by its energy spectrum. We prove that analogous statements also hold true for Kleinian surface groups.