Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

Dai, Song; Li, Shuyu

doi:10.1112/plms.12431

Cited by 4 publications

(6 citation statements)

References 44 publications

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“…Remark 2.3. Dai and the author in [4] prove a generalized theorem of Lemma 2.2 and give an independent proof of Lemma 2.2 as a byproduct.…”

Section: Algebraic Inequalitiesmentioning

confidence: 99%

“…Lemma 4.3. (Dai-Li [4], Lemma 4.6 and 4.7) Let j be a Fuchsian representation and f j be the associated harmonic map.…”

Section: Nilpotent Higgs Bundles On Riemann Surfacesmentioning

confidence: 99%

“…Lemma 4.4. (Dai-Li [4], Section 5) Suppose a ρ-equivariant harmonic map f : Σ → SL(n, C)/SU (n) is a totally geodesic immersion and the curvature of its image is a negative constant −c. Then there is a Fuchsian representation j and a representation µ π : π 1 (Σ) → G π such that the corresponding j-equivariant harmonic map f j : Σ → H 2 , a partition π ∈ P n and an element x ∈ SL(n, C) such that…”

Section: Nilpotent Higgs Bundles On Riemann Surfacesmentioning

confidence: 99%

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Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli

2020

Preprint

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We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi-Yau moduli.Corollary 1.2. For any nilpotent polystable SL(n, C)-Higgs bundle over Σ, the corresponding representation ρ satisifies l ρ ≤ α • l τn•j Σ for some positive constant α < 1, unless P(ρ) = P(τ n • j Σ ) .Remark 1.3. The nilpotent cone consisting of nilpotent Higgs bundles is the fiber at 0 of the Hitchin fibration from the moduli space of Higgs bundles to the vector space n j=2 H 0 (Σ, K i ). In [4], Dai and the author generalize Corollary 1.2 to Higgs bundles in Hitchin fibers which contain n-Fuchsian Higgs bundles.

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“…Remark 2.3. Dai and the author in [4] prove a generalized theorem of Lemma 2.2 and give an independent proof of Lemma 2.2 as a byproduct.…”

Section: Algebraic Inequalitiesmentioning

confidence: 99%

“…Lemma 4.3. (Dai-Li [4], Lemma 4.6 and 4.7) Let j be a Fuchsian representation and f j be the associated harmonic map.…”

Section: Nilpotent Higgs Bundles On Riemann Surfacesmentioning

confidence: 99%

Section: Nilpotent Higgs Bundles On Riemann Surfacesmentioning

confidence: 99%

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Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli

2020

Preprint

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“…Finally, it would be interesting to relate Corollary 5.4, or an analog of it, to the recent work by Dai–Li [22] studying the translation lengths on the symmetric space of

{{\mathsf {PSL}}}_d(\mathbb {C})

, when one deforms a Fuchsian representation along its Hitchin fiber.…”

Section: Pluriharmonicity Of Length Functions and Its Consequencesmentioning

confidence: 99%

Hessian of Hausdorff dimension on purely imaginary directions

Bridgeman

Pozzetti

Sambarino

et al. 2022

Bulletin of London Math Soc

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Bridgeman-Taylor (Math. Ann. 341 (2008), [927][928][929][930][931][932][933][934][935][936][937][938][939][940][941][942][943] and McMullen (Invent. Math. 173 (2008), 365-425) showed that the Weil-Petersson metric on Teichmüller space can be realized by looking at the infinitesimal change of the Hausdorff dimension of certain quasi-Fuchsian deformations. In this article, we give a similar geometric interpretation of the spectral gap pressure metric introduced by Bridgeman-Canary-Labourie-Sambarino (Geom. Dedicata 192 (2018), 57-86) on the Hitchin component for 𝖯𝖲𝖫 𝑑 (ℝ). More generally, we investigate the Hessian of the Hausdorff dimension as a function on the space of (1,1,2)-hyperconvex representations, a class introduced in (J. reine angew. Math. 774 (2021), 1-51) which includes small complex deformations of Hitchin representations and of Θ-positive representations. As another application, we prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ → 𝖯𝖮(𝑛, 1) → 𝖯𝖴(𝑛, 1) is positive definite when Γ is co-compact in 𝖯𝖮(𝑛, 1) (unless 𝑛 = 2 and the deformation is tangent to 𝔛(Γ, 𝖯𝖮(2, 1))).M S C ( 2 0 2 0 ) 22E40 (primary), 37Dxx (secondary)

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“…Marché and Wolff use domination to answer a question of Bowditch and resolve the Goldman conjecture in genus 2 [32]. Dai-Li study domination for higher rank Hitchin representations into PSL(n, C) [9]. It would be interesting to see if the almost strict condition generalizes meaningfully to higher rank.…”

Section: Recent Workmentioning

confidence: 99%

Almost strict domination and anti-de Sitter 3-manifolds

Sagman¹

2021

Preprint

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We define a notion called almost strict domination for pairs of representations ρ1 : π1(Sg,n) → PSL(2, R), ρ2 : π1(Sg,n) → G, where G is the isometry group of a Hadamard manifold (X, ν). We prove that ρ1 almost strictly dominates ρ2 if and only if one can find a (ρ1, ρ2)-equivariant spacelike maximal surface in the pseudo-Riemannian manifold (H × X, σ ⊕ (−ν)), which is unique up to fixing certain parameters. Adapting a construction of Tholozan, we construct all such representations and parametrize the deformation space.When (X, ν) = (H, σ), we establish a correspondence between a family of maps into (H × H, σ ⊕ (−σ)) and anti-de Sitter structures on Seifert-fibered 3-manifolds. In this way, we show that an almost strictly dominating pair gives rise to a Kleinian anti-de Sitter 3-manifold. Moreover, we can extract geometric information about the 3-manifold using maximal surfaces and associated Higgs bundles. The general results on maximal surfaces allow us to parametrize a deformation space of anti-de Sitter 3-manifolds as a union of components in a PSL(2, R) × PSL(2, R) relative representation variety, analogous to classical Teichmüller spaces.

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Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

Cited by 4 publications

References 44 publications

Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli

Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli

Hessian of Hausdorff dimension on purely imaginary directions

Almost strict domination and anti-de Sitter 3-manifolds

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