The dual Bonahon–Schläfli formula

Mazzoli, Filippo

doi:10.2140/agt.2021.21.279

Cited by 4 publications

(9 citation statements)

References 16 publications

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“…1.3). On the other hand, the first order variation of the dual volume, via an application of the Bonahon-Schläfli formula is expressed as ( [45,55]):…”

Section: Analogy Between Bending Laminations and Measured Foliations ...mentioning

confidence: 99%

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

2023

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The main subject of this thesis are a certain class of hyperbolic 3-manifolds called quasi-Fuchsian manifold. Given an orientied, closed hyperbolic surface S, these manifolds are homeomorphic to S × R. We study two questions regarding them: one is on measured foliations at infinity and the other is on foliation by constant mean curvature surfaces.Measured foliations at infinity of quasi-Fuchsian manifolds are a natural analog at infinity to the measured bending laminations on the boundary of its convex core. Given a pair of measured foliations (F+, F−) which fill a closed hyperbolic surface S and are arational, we prove that for t > 0 sufficiently small tF+ and tF− can be uniquely realised as the measured foliations at infinity of a quasi-Fuchsian manifold homeomorphic to S × R, which is sufficiently close to the Fuchsian locus. The proof is based on that of Bonahon in [5] which shows that a quasi-Fuchsian manifold close to the Fuchsian locus can be uniquely determined by the data of filling measured bending laminations on the boundary of its convex core. Finally, we interpret the result in half-pipe geometry.For the second part of the thesis we deal with a conjecture due to Thurston asks if almost-Fuchsian manifolds admit a foliation by CMC surfaces. Here, almost-Fuchsian manifolds are defined as quasi-Fuchsian manifolds which contain a unique minimal surface with principal curvatures in (−1, 1) and it is known that in general, quasi-Fuchsian manifolds are not foliated by surfaces of constant mean curvature (CMC) although their ends are. However, we prove that almost-Fuchsian manifolds which are sufficiently close to being Fuchsian are indeed monotoni-cally foliated by surfaces of constant mean curvature. This work is in collaboration with Filippo Mazzoli and Andrea Seppi.1. The Codazzi equation, d ∇ h = 0, where ∇ is the Levi-Civita connection of g.

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“…1.3). On the other hand, the first order variation of the dual volume, via an application of the Bonahon-Schläfli formula is expressed as ( [45,55]):…”

Section: Analogy Between Bending Laminations and Measured Foliations ...mentioning

confidence: 99%

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

2023

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“…2 `m. /: In [29], a first-order variation formula for the function V C over QD.M / is studied, called the dual Bonahon-Schläfli formula,…”

Section: Dual Volumementioning

confidence: 99%

“…0 / is the length of the bending measure of @CM 0 . As a consequence of the variation formulae of V C (see Bonahon [4]) and of V C (see Mazzoli [29] and see also Krasnov and Schlenker [20]), we will see in Corollary 4.1 that the multiplicative constant 1 2 is optimal, and is realized near the Fuchsian locus. Theorem A is to the dual volume as the following result of Bridgeman, Brock and Bromberg is to the renormalized volume: Theorem 3.11] For every convex cocompact hyperbolic 3-manifold M with incompressible boundary, inf…”

Section: Introductionmentioning

confidence: 99%

“…@ 1 M /. In order to study the dual volume function, the analogy between the variation formula of the renormalized volume (see the work of Krasnov and Schlenker [19,Lemma 5.8], or Section 1.6) and the dual Bonahon-Schläfli formula [29] would tempt us to consider V C as a function of the Teichmüller space T . @CM/, seen as the deformation space of hyperbolic structures on the boundary of the convex core of M .…”

Section: Introductionmentioning

confidence: 99%

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The infimum of the dual volume of convex cocompact hyperbolic 3–manifolds

Mazzoli

2023

Geom. Topol.

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The infimum of the dual volume of convex cocompact hyperbolic 3-manifolds FILIPPO MAZZOLIWe show that the infimum of the dual volume of the convex core of a convex cocompact hyperbolic 3-manifold with incompressible boundary coincides with the infimum of the Riemannian volume of its convex core, as we vary the geometry by quasi-isometric deformations. We deduce a linear lower bound of the volume of the convex core of a quasi-Fuchsian manifold in terms of the length of its bending measured lamination, with optimal multiplicative constant. 30F40; 52A15, 57M50

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“…One of the principal motivations to consider questions 1.3 and 1.4 is the remarkable similarity between the variational formulae for the dual volume V * C (M ) of the convex core of M (see [27]) and the renormalised volume V R of M , (see [26]). Suppose for 0 ≤ t < , we have a differentiable path of quasi-Fuchsian structures on M given by t → M t , then the first order variation of the dual volume, via an application of the Bonahon-Schläfli formula, is expressed as ( [27], [31]):…”

Section: Analogy With Measured Bending Laminations On the Convex Corementioning

confidence: 99%

Measured foliations at infinity of quasi-Fuchsian manifolds near the Fuchsian locus

Choudhury¹

2021

Preprint

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Given a pair of measured foliations (F + , F − ) which fill a closed hyperbolic surface S, we show that for t > 0 sufficiently small, tF + and tF − can be uniquely realised as the measured foliations at infinity of a quasi-Fuchsian hyperbolic 3-manifold homeomorphic to S × R, which is in a suitably small neighbourhood of the Fuchsian locus. This is parallel to a theorem of Bonahon that partially answers a conjecture of Thurston by proving that a quasi-Fuchsian manifold close to being Fuchsian can be uniquely determined by the data of measured bending laminations on the boundary of its convex core. We also give an interpretation of our result in half-pipe geometry using intermediate steps in the proof.

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The dual Bonahon–Schläfli formula

Cited by 4 publications

References 16 publications

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

The infimum of the dual volume of convex cocompact hyperbolic 3–manifolds

Measured foliations at infinity of quasi-Fuchsian manifolds near the Fuchsian locus

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