Joseph D. Masters, Xingru Zhang
We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.
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In this paper we give the characterization of Fuchsian groups acting on quaternionic hyperbolic 2-space.
We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
On totally hyperbolic non-Fuchsian type-preserving representations
本文研究穿孔曲面基本群到PSL(2,ℝ)的保型表示，识别出一类全双曲非Fuchsian表示，否定了Bowditch的相关问题，刻画了其欧拉类、共轭类分布及子曲面限制性质。
We prove a volume-rigidity theorem for fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom(H^n). Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of its fundamental group into Isom(H^n), 3 <= k <= n, is less than the volume of M, and the volume is maximal if and only if the representation is discrete, faithful and ``k-fuchsian''.
We indicate a C-Fuchsian counter-example to the result with the above title announced at http://www.maths.dur.ac.uk/events/Meetings/LMS/2011/GAL11/program.pdf and prove a stronger statement.
The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.
In this paper, we will determine the topological types of hyperbolic 3-anifolds H^3/G such that G is a geometric limit of any algebraically convergent sequence of quasi-Fuchsian groups.
We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the \textit{geodesic covers} of the Fuchsian group or the hyperbolic surface. The paper contributes to showing that finiteness of geodesic covers is basically another characterization of geometrically finiteness. The resolution of geometrically finite case is based on Shimizu's lemma.
We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3…
Measured foliations at infinity and constant mean curvature surface in quasi-Fuchsian manifolds close to the Fuchsian locus
本文研究接近Fuchsian轨迹的拟Fuchsian流形，证明小尺度填满型无理测度叶状结构可唯一实现为无穷远测度叶状结构，并证明近Fuchsian流形可由CMC曲面单调叶状化。
We construct infinitely many noncommensurable non-cocompact Fuchsian groups $Δ$ of finite covolume sitting in PSL(2,Q) so that the set of hyperbolic fixed points of $Δ$ will contain a given finite collection of elements in the boundary of the hyperbolic plane.
In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two frames with a path of frames in a prescribed relative homology class of the frame bundle. The existence result is applied to show that every uniform lattice of $\mathrm{PSL}(2,\mathbb{C})$ admits an exhausting nested sequence of sublattices with exponential homological torsion growth…
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.
Surface Quotients of Right-Angled Hyperbolic Buildings
本文研究 Fuchsian 建筑的曲面商，证明一大类建筑存在自同构群离散子群使其商为无边界紧致曲面，并给出基于建筑对称性的必要条件。
This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise's theorem that the fundamental group of M acts freely and cocompactly on a CAT(0) cube complex.
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…
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. We show that given a pair of arational measured foliations $(\mathsf{F}_{+},\mathsf{F}_{-})$ which fill a closed hyperbolic surface $S$, for $t>0$ sufficiently small, $t\mathsf{F}_{+}$ and $t\mathsf{F}_{-}$ can be uniquely realised as the measured foliations at infinity of a quasi-Fuchsian manifold homeomorphic to $S\times \mathbb{R}$, which is sufficiently close to the Fuchsian locus. Here arationality means that the corresponding…
We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.
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…
Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class.
Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $π_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $π_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact $π_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques develo…
Poisson bracket of trace functions on the Fuchsian locus and Wolpert's formulas
该研究针对双曲曲面上莫比乌斯变换乘积的迹，开发了基于SL₂(ℝ)矩阵归一化的系统计算方法，结合Goldman的辛形式描述计算相关迹函数的泊松括号，应用中得到Wolpert的余弦与正弦公式。
Cheeger constant of convex co-compact hyperbolic 3-manifolds
研究拟Fuchsian或无柱型凸余紧双曲3-流形空间中的Cheeger等周常数泛函，证明其全局最大值在Fuchsian轨迹处唯一取得。
For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizer of a reprentative of each group orbit of mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some of the triangle groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.
Thurston and Calegari-Dunfield showed that the fundamental group of some tautly foliated hyperbolic 3-manifold acts on the circle in a distinctive way that the action preserves some structure of S^1, so-called a circle lamination. Indeed, a large class of Kleinian groups acts on the circle preserving a circle lamination. In this paper, we are concerned with the converse problem that a group acting on the circle with at least two invariant circle laminations is Kleinian. We prove that a subgroup of the orientation preserving circle homeomorphism group is a Fuchsian group whose quotient orbifold…
Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold
研究了封闭双曲3流形中渐近弗uchsian面的极限性质，证明其测度极限由双曲测度与完全测地面测度的凸组合构成。
Bending parameterization of geometrically finite hyperbolic manifolds
本文证明非 Fuchs 型几何有限双曲结构由弯曲叶状结构唯一确定，弯曲映射为同胚，推广了凸余紧情形，并建立了边界纤维可收缩性判据。
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。
We study global aspects of the mean curvature flow of non-separating hypersurfaces $S$ in closed manifolds. For instance, if $S$ has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal hypersurface $Γ$. We prove a similar result for the flow with surgery in dimension 2. As an application we show the existence of monotone incompressible isotopies in manifolds with negative curvature. Combining this result with min-max theory, we show that quasi-Fuchsian and hyperbolic $3$-manifolds fibered over $\mathrm{S}^1$ admit smooth entire foliations who…
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and the real line. In particular, we prove that there exists a large class of closed initial surfaces, as geodesic graphs over the totally geodesic surface $Σ$, such that the mean curvature …
A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.
Let $S=Γ\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $Γ\le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any generating set $\mathfrak S$ of $Γ$. When sampling by an $n$-step random walk in $π_1(S) \cong Γ$ with each step given by an element in $\mathfrak S$, the subset of this sampled set comprised of hyperbolic elements approaches full measure as $n\to \infty$, and for this subset, the distribution of geometric lengths obeys a Law of Large Numbers, Central Limit Theorem, Large Deviations Principle, and Local Limit…
Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends
本文针对具有 Fuchsian 端的高维凸余紧双曲流形，借助规避定理克服中间群闭轨道聚集障碍，建立了幂幺子群作用的 Ratner 型轨道闭包定理，并推出水平球面、测地平面闭包及稠密性结果。
We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.
Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold $M$ is bounded by an explicit constant, depending only on the topology of $M$, times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of the convex core of $M$.
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。
Geometry and arithmetic of semi-arithmetic Fuchsian groups
本文引入了半算术Fuchsian群的新几何不变量stretch，基于Riemannian质心概念，并利用算术Margulis引理证明了在算术维数、stretch和余面积有界时仅有有限多个共轭类，无界时则存在无限序列。
We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and…
Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus
本文证明了在Fuchsian轨迹附近，具有严格凸类空边界（光滑或多面体）的整体双曲紧致反德西特(2+1)时空由边界诱导度量唯一确定。
We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.
Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends toward a pair of projectively measured laminations that bind the surface, there is a convergent subsequence. This preprint also analyzes the quasi-isometric geometry of quasi-Fuchsian 3-manifolds. This eprint is based on a 1986 preprint, which was refereed and accepted for publication, but which I neglected to correct and return. The referee's corrections have…
Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps …
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。
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。
According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to…
Quantitative FUP and spectral gap for quasi-Fuchsian group
本文通过量化高维分形不确定性原理，为由准费米群产生的超曲面3流形提供了显式的本质谱间隙，并改进了Tao 2025的结果。
We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameterise a neighborhood of the R-Fuchsian representations in the representation space by pairs consisting of a point in Teichmuller space and a small cubic differential. By constructing a fundamental domain, we show these representations are complex-hyperbolic quasi-Fuchsian, thus rec…
This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite or that it has finitely generated fundamental group.
Fields of definition for triangle groups as Fuchsian groups
本文研究紧双曲三角形群在迹域上的定义域问题，证明恰好有十一个这样的群共轭于PSL₂(K)的子群，且对任意全实域L不存在额外情形，从而回答了Waterman–Machlachlan的问题并证实了McMullen的五个相关猜想。
We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $π$: any first-order deformation changes either one of those angles or the conformal structure at infinity, with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure.
We describe a family of representations of $π_1(Σ)$ in PU(2,1), where $Σ$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of $Σ$. We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichmüller space. We identify within this family new examples of discrete, faithful and type-preserving representations of $π_1(Σ)$. In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of $Σ$ in the PU(2,1)-representati…
We study geodesics on a planar Riemann surface of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing the surface.
Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in $S^4$
该研究针对四维球面中二维环面的双曲链环补空间，证明其内部不存在闭的嵌入全测地双曲3-流形，为高维双曲流形的几何结构研究提供了新结论。
Bending parameterization of one-sided degenerate Kleinian surface groups
本文针对出现在Bers切片边界中的单侧退化非拟富克斯流形，证明其由退化端的端结构（含抛物轨迹及补集分支的终结叶状结构/共形结构）与另一端的弯曲叶状结构唯一确定，拓展了拟富克斯流形的相关唯一性结果。
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We also generalize the main result for relatively hyperbolic groups.
Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.
Connectivity in the space of framed hyperbolic 3-manifolds
该研究证明了带框架的无限体积双曲3维流形空间$\mathcal{H}_\infty$连通但不路径连通，通过构造稠密框架集和非 tame 流形族，揭示了其拓扑结构特性。
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound of the length of an $n^{th}$ shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$. Our works complement the work of Lakeland and Leininger \cite{Christopher} on the upper bound of systole length.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to a standard $S^2 \times S^2$.
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