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Journal Article

The energy spectrum of metrics on surfaces


Slegers,  Ivo
Max Planck Institute for Mathematics, Max Planck Society;

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Slegers, I. (2022). The energy spectrum of metrics on surfaces. Geometriae Dedicata, 216(4): 47. doi:10.1007/s10711-022-00704-8.

Cite as: https://hdl.handle.net/21.11116/0000-000A-A954-9
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.