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Journal Article

#### The energy spectrum of metrics on surfaces

##### External Resource

https://doi.org/10.1007/s10711-022-00704-8

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##### Fulltext (public)

Slegers_The energy spectrum of metrics on surfaces_2022.pdf

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##### Citation

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

##### 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.

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.