Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations

Slegers, Ivo

doi:10.48550/arxiv.2011.03936

Cited by 2 publications

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“…Furthermore, he made the conjecture that it has a unique minimum. The author showed in [20] that in this same setting the energy spectrum is strictly plurisubharmonic.…”

Section: Introductionmentioning

confidence: 98%

The energy spectrum of metrics on surfaces

Slegers

2022

Geom Dedicata

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Let $$(N,\rho )$$ ( N , ρ ) be a Riemannian manifold, S a surface of genus at least two and let $$f:S \rightarrow N$$ f : S → N be a continuous map. We consider the energy spectrum of $$(N,\rho )$$ ( N , ρ ) (and f) which assigns to each point $$[J]\in \mathcal {T}(S)$$ [ J ] ∈ T ( S ) in the Teichmüller space of S the infimum of the Dirichlet energies of all maps $$(S,J)\rightarrow (N,\rho )$$ ( S , J ) → ( N , ρ ) homotopic to f. We study the relation between the energy spectrum and the simple length spectrum. Our main result is that if $$N=S$$ N = S , $$f={{\,\mathrm{id}\,}}$$ f = id and $$\rho $$ ρ 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.

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“…Furthermore, he made the conjecture that it has a unique minimum. The author showed in [20] that in this same setting the energy spectrum is strictly plurisubharmonic.…”

Section: Introductionmentioning

confidence: 98%

The energy spectrum of metrics on surfaces

Slegers

2022

Geom Dedicata

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, he made the conjecture that it has a unique minimum. The author showed in [Sle20] that in this same setting the energy spectrum is strictly plurisubharmonic.…”

Section: Introductionmentioning

confidence: 98%

The energy spectrum of metrics on surfaces

Slegers¹

2021

Preprint

Self Cite

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Let (N, ρ) be a Riemannian manifold, S a surface of genus at least two and let f : S → N be a continuous map. We consider the energy spectrum of (N, ρ) (and f ) which assigns to each point [J] ∈ T (S) in the Teichmüller space of S the infimum of the Dirichlet energies of all maps (S, J) → (N, ρ) 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 ρ 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.

show abstract

Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations

Cited by 2 publications

References 0 publications

The energy spectrum of metrics on surfaces

The energy spectrum of metrics on surfaces

The energy spectrum of metrics on surfaces

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