Francis Bonahon
We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.
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Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.
We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We …
An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual boundary of hyperbolic 3-space. This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal geometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups.
Beyond almost Fuchsian space
本文研究了弱几乎 Fuchsian 流形的几何有限性，构造了其空间的紧化，并据此证明了凸核体积与极限集维数的一致上界及主曲率间隙定理。
Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1, 1). In such a manifold, the minimal surface is unique and embedded, hence one can parametrize these hyperbolic three-manifolds by their minimal surfaces. In this paper we obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold M in terms of the data on the minimal surface. In particular, we obtain an upper bound for the…
An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by their minimal surfaces. We prove that any closed surface which is a graph over any fixed surface of small principal curvatures can be deformed into the minimal surface via the mean curvature flow. We also obtain an upper bound for the hyperbolic volume of the convex core of M, a…
On Almost-Fuchsian submanifodls of Hadamard spaces and the asymptotic Plateau problem
本文研究了负曲率空间中具有小曲率的极小子流形，证明了在曲率负被夹的Hadamard空间中，第二基本形受限的完备极小子流形能使渐近Plateau问题对一类无穷远球面唯一可解。
Almost-fuchsian structures on disk bundles over a surface
本文证明了在亏格足够大的曲面上，存在凸余紧表示使得相关的双曲4-流形为度数为d的圆盘丛，并给出了非富克斯形变的广义几乎富克斯表示的例子。
We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds.
Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds
本文研究近 Fuchs 流形的 CMC 叶状化猜想，通过证明一类流形中修正平均曲率流的长时间存在与收敛，证实该子类满足 Thurston 猜想。
Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces
本文证明靠近 Fuchsian 轨迹的 quasi-Fuchsian 双曲三流形具有唯一的单调常平均曲率曲面叶状结构，为 Thurston 相关猜想提供邻近情形的结论。
Weakly almost-Fuchsian manifolds are nearly-Fuchsian
本文证明含主曲率在[-1,1]的闭极小曲面的双曲三维流形必含主曲率在(-1,1)的邻近曲面，进而推出完备同胚于S×R的此类流形为拟Fuchsian，并否证了2000年代的一个猜想。
On a convexity property of the space of almost fuchsian immersions
本文研究紧致黎曼曲面上几乎福克斯极小浸入的Hopf微分空间，证明外曲率是Hopf微分的凹函数，从而该集合为凸子集，并给出非等变情形下集合大小的上下界。
Almost-Fuchsian representations in PU(2,1)
本文研究了PU(2,1)中非最大Toledo不变量的凸-紧致表示，通过构造全纯且几乎全等距的极小曲面，展示了高 genus 下的构造方法及主要贡献。
Donaldon constructed a hyperkähler moduli space $\mathcal{M}$ associated to a closed oriented surface $Σ$ with $\textrm{genus}(Σ) \geq 2$. This embeds naturally into the cotangent bundle $T^*\mathcal{T}(Σ)$ of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to $Σ$. The later is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bun…
An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichmüller space is represented as a path $γ$, we show that $γ$ joins the conformal structures of the two components of the conformal boundary of $M$. Moreover, we obtain an upper bound for the Teichmüller distance between any two points on $γ$, in particular, the Teichmüller distance between the two components of the conformal boundary of $M$, in term…
It is known that the almost-Kaehler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out entire connected components in the moduli space. Our construction hinges on an unexpected link between harmonic functions on certain hyperbolic 3-manifolds and self-dual harmonic 2-forms on associated 4-manifolds.
Let $Γ$ be a finitely generated group, and let $\op{Rep}(Γ, \SO(2,n))$ be the moduli space of representations of $Γ$ into $\SO(2,n)$ ($n \geq 2$). An element $ρ: Γ\to \SO(2,n)$ of $\op{Rep}(Γ, \SO(2,n))$ is \textit{quasi-Fuchsian} if it is faithful, discrete, preserves an acausal subset in the conformal boundary $\Ein_n$ of the anti-de Sitter space; and if the associated globally hyperbolic anti-de Sitter space is spatially compact - a particular case is the case of \textit{Fuchsian representations}, i.e. composition of a faithfull, discrete and cocompact representation $ρ_f: Γ\to \SO(1,n)$ an…
Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations
本文引入双复双曲空间中的复拉格朗日极小曲面与双复希格斯丛，推广了多类极小曲面与表示的相关研究，给出$\rm SL(3,\boldsymbol{\rm C})$-拟富克斯表示的参数化并证明其空间具有双复结构。
Entropy rigidity for cusped Hitchin representations
本文针对几何有限Fuchsian群的带尖点Hitchin表示建立熵刚性定理，引入(1,1,2)-超横截群类与射影可见群横截表示理论，推广了多个相关已有结论。
In a previous paper by Deroin-Tholozan, the authors construct a map $\mathbfΨ_ρ$ from the Teichmüller space of $S$ to itself and prove that, when $M$ has sectional curvature $\leq -1$, the image of $\mathbfΨ_ρ$ lies (almost always) in the domain $\mathrm{Dom}(ρ)$ of Fuchsian representations stricly dominating $ρ$. Here we prove that $\mathbfΨ_ρ: \mathrm{Teich}(S) \to \mathrm{Dom}(ρ)$ is a homeomorphism. As a consequence, we are able to describe the topology of the deformation space of anti-de Sitter structures on closed 3-manifolds.
Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by the Schwarzian parametrization. This is done in restriction to the moduli space of almost-Fuchsian structures by involving a notion of renormalized volume, used to relate the geometry of a minimal surface in a hyperbolic 3-manifold to the geometry of its ideal conformal boundary.
Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$
该研究构造了$\boldsymbol{\rm PSL}(2,\boldsymbol{\rm C})$中极小浸入的形变空间，定义了映射类群不变的复超凯勒结构，其限制对应多种已知几何结构且轨迹全测地，还探讨了相关$\boldsymbol{\rm C}^*$-作用的哈密顿性质。
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