On the construction of minimal foliations by hyperbolic surfaces on 3-manifolds

Cuesta, Fernando Alcalde; Dal’Bo, Françoise; Martínez, Matilde María Olarte; Verjovsky, Alberto

doi:10.48550/arxiv.1611.09833

Cited by 2 publications

(5 citation statements)

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“…The free homotopy class of O (w ) is the conjugacy class of some element γ ∈ π 1 (Σ) and the holonomy map τ w over O (w ) is conjugated to hol(γ) −1 . In particular it lies in PSL 2 (R) and, since O (v) is closed, has a periodic point in RP 1 : it has to be conjugated to a rational rotation, a parabolic or hyperbolic element. In the first case the holonomy over O (w ) is conjugated to an isometry and we clearly have λ ⋔ (µ v ) = 0.…”

Section: Domination Of Representations Implies Partial Hyperbolicitymentioning

confidence: 99%

“…We shall define in Appendix what is a foliation by hyperbolic surfaces. Many examples may be found in [1]. Let us emphasize that every two-dimensional foliation without transverse invariant measure must be a foliation by hyperbolic surfaces: see Proposition 6.3.…”

Section: Introductionmentioning

confidence: 99%

“…1 -bundle with projective holonomy over a closed surface Σ endowed with a hyerbolic metric m. Endow M with an admissible Riemannian metric. Let ρ : π 1 (Σ) → PSL 2 (R) denote a Fuchsian representation associated to m and hol : π 1 (Σ) → PSL 2 (R) denote the holonomy representation of F .…”

mentioning

confidence: 99%

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Physical measures for the geodesic flow tangent to a transversally conformal foliation

Alvarez¹,

Yang²

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider a transversally conformal foliation F of a closed manifold M endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either there is a transverse holonomy-invariant measure for F , or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin cover a set full for the Lebesgue measure. We also give necessary and sufficient conditions for the foliated geodesic flow to be partially hyperbolic in the case where the foliation is transverse to a projective circle bundle over a closed hyperbolic surface.

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Section: Domination Of Representations Implies Partial Hyperbolicitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Physical measures for the geodesic flow tangent to a transversally conformal foliation

Alvarez¹,

Yang²

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…Direct computations with linear coefficients give a partial result on H * π (M ), and we describe the dimensions of the cohomology groups and its generators. In the following tables ( 5), (6), and (7)…”

Section: A Remark On Poisson Cohomologymentioning

confidence: 99%

“…The relationship between foliation theory and other topics of 3-manifolds is still being explored (e.g. [2,7,10,24]). Various analytic, geometric, and topological complications arise when singularities are allowed to exist in a foliation.…”

Section: Introductionmentioning

confidence: 99%

On Bott-Morse Foliations and their Poisson structures in dimension three

Evangelista-Alvarado¹,

Suárez–Serrato²,

Orozco³

et al. 2019

jsing

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We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.This section follows notations used in [20] and [21], where Bott-Morse foliations on dimension 3 were described.Let M m be a closed, orientable, smooth manifold of dimension m, for m ≥ 3. Let F be a codimension-one smooth foliation with singularities on M . Denote by Sing(F) the set of singular points of F.

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On the construction of minimal foliations by hyperbolic surfaces on 3-manifolds

Cited by 2 publications

References 0 publications

Physical measures for the geodesic flow tangent to a transversally conformal foliation

Physical measures for the geodesic flow tangent to a transversally conformal foliation

On Bott-Morse Foliations and their Poisson structures in dimension three

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