Bruno Martelli, Stefano Riolo, Leone Slavich
We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.
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We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cut-and-paste constructions of symplectic 4-manifolds.
In this paper we show that a given set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3-manifolds with that specified length spectrum, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.
This paper investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume of the doubled manifold DM, and this inequality is sharp. This paper proves that the inequality is in fact sharp in every pleating variety of AH(M).
A version of a conjecture of McMullen is as follows: Given a hyperbolizable 3-manifold M with incompressible boundary, there exists a uniform constant K such that if N is a hyperbolic 3-manifold homeomorphic to the interior of M, then the injectivity radius based at points in the convex core of N is bounded above by K. This conjecture suggests that convex cores are uniformly congested. We will give a proof in the case when M is an I-bundle over a closed surface, taking into account the possibility of cusps.
Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces c…
New family of hyperbolic knots whose Upsilon invariants are convex
针对Borodzik和Hedden提出的Upsilon不变量为凸的纽结分类问题，本文采用$(1,1)$-纽结的组合方法计算纽结Floer复形，构造出无穷多互不协边且Upsilon不变量为凸的双曲纽结，丰富了该类纽结的实例。
Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.
We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold $M_i$ $δ$-hyperbolic and with Margulis constants independent of $i$. By taking the convex hull in the cover of $M_i$ corresponding the core, we show that there exists an exiting sequence of surfaces $Σ_i$. We drill out the covers of $M_i$ by a core $C$ a…
Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles $A_C$ obtained by restricting to {\em non-obtuse} angles is a convex polytope. In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the space of dihedral angles of compact hyperbolic tetrahedra,…
We describe in this paper a geometric construction in the projective p-adic plane that gives, together with a suitable notion of p-adic convexity, some open subsets of P 2 .Q p / naturally endowed with a "Hilbert" distance and a transitive action of PGL.2; Q p / by isometries. ese open sets are natural analogues of the hyperbolic disc, more precisely of Klein's projective model. But, unlike the real case, there is not only one such hyperbolic disc. Indeed, we nd three of them if p is odd (and seven if p D 2). Let us stress out that neither the usual notion of convexity nor that of connectednes…
Convex co-compact hyperbolic manifolds are determined by their pleating lamination
本文证明凸余紧三维双曲流形可由其凸核边界上的褶皱测度层合唯一确定，解决了该类流形的刚性问题。
Convex Hull Volumes in Hyperbolic 3-Space
研究双曲三维空间中闭集的几何条件，证明其凸包体积无限，回答Danny Calegari问题，并刻画平面自相似集的几何特性。
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。
We reinterpret the renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in $\mathbb{H}^3$. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in $\mathbb{H}^3$ or in cyclic quotients of $\mathbb{H}^3$.
The infimum of the dual volume of convex co-compact hyperbolic $3$-manifolds
本文证明凸余紧双曲3-流形凸核的对偶体积与黎曼体积下确界一致，并推出拟富克斯流形体积关于弯曲层合长度的最优线性下界。
We consider globally hyperbolic maximal anti de Sitter 3-manifolds $M$ with a closed Cauchy surface $S$ of genus greater than one and prove that any pair of hyperbolic metrics on $S$ can be realized as the boundary metrics of the convex core of a maximal globally hyperbolic anti de Sitter 3-manifold structure on $M$. This answers the existence part of a question of Mess about the unique realization of such metrics. Our theorem has a nice formulation purely in terms of 2-dimensional Teichmüller theory and earthquakes.
We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the Poincare metric when the domain is uniformly perfect. We also establish Marden and Markovic's conjecture that a hyperbolic domain is uniformly perfect if and only if the nearest point retraction is Lipschitz with respect to the Poincare metric.
Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space
本文证明Koebe圆域猜想等价于零亏格完备双曲曲面可实现为圆域补集双曲凸包边界的Weyl型问题，并借助He-Schramm工作证明单连通非紧多面体曲面离散共形等价于复平面或单位圆盘。
Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere
论文通过两个例子展示实超曲面的等距群中凸紧致子群的极限集为庞特里亚金球，并探讨其与Menger曲线的关联。
Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space H^{p,q-1} in place of the Riema…
Convex co-compact actions of relatively hyperbolic groups
本文研究了实射影空间真凸域上凸余紧作用的离散群，基于凸域几何性质建立了群相对双曲的充要条件，回答了 Danciger-Guéritaud-Kassel 的问题。
The structure of relatively hyperbolic groups in convex real projective geometry
本文研究了在凸实射影几何中相对超几何群的结构，通过凸紧致作用和不变凸子集的特征化，建立了与CAT(0)空间结果的类比，并给出了边界之间的等距同胚。
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y) with X and Y in its boundary is comparable to the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the Weil-Petersson distance with the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian as well as a new finiteness …
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamated free product. Constructions of infinite volume hyperbolic n-manifolds are described by gluing lower dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multi…
Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist's theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenev…
A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space
本文研究了双曲三维空间中闭合凸集的刚性问题，证明了在极限点集具有零长度时，其边界度量可确定该集，且该条件最优，类比复分析中的Painlevé定理。
Hyperbolic 3-manifolds with boundary of polyhedral type
本文研究具有双曲内部和凸边界的紧致3-流形，证明边界上凸双曲锥度量可由弯曲度量实现且可控多面体实现唯一，并得出凸余紧度量的全局与无穷小刚性，推进Thurston猜想。
Let $(M, \partial M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature $K>-1$, and that the third fundamental forms of $\dr M$ are exactly the metrics with curvature $K<1$, for which contractible closed geodesics have length $L>2π$. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the met…
Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space
研究双曲三维空间中非紧凸集的刚性，证明若两个非紧闭凸子集的理想边界为圆型闭集且连通分支可数，则边界间的内蕴等距可延拓为整体等距，推广了Luo-Luo-Rao的刚性定理，并建立了Weyl问题的唯一性结果。
In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $Σ$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geq 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.
Transition of convex core doubles from hyperbolic to Anti-de sitter geometry
本文提出一种构造方法，研究曲面 $Σ\times\mathbb{S}^1$ 上凸核倍流形在弯曲叶状结构均匀坍缩至零时，从双曲几何经Half-pipe几何过渡到Anti-de Sitter几何，并沿链产生锥奇点。
It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $\langle H, R\rangle\cong H\ast R$ with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in $G$. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits…
A version of a conjecture of McMullen is as follows: Given a hyperbolizable 3-manifold M with incompressible boundary, there exists a uniform constant K such that if N is a hyperbolic 3-manifold homeomorphic to the interior of M, then the injectivity radius based at points in the convex core of N is bounded above by K. This conjecture suggests that convex cores are uniformly congested. In previous work, the author has proven the conjecture for $I$-bundles over a closed surface, taking into account the possibility of cusps. In this paper, we establish the conjecture in the case that M is a book…
$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds
本文研究了凸余紧双曲3-流形钻孔短测地线后边界复射影结构的变化，给出了不依赖于边界单射半径的$L^2$-界，证明其变化受普适常数乘以测地线长度平方根的约束。
We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.
Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space
本文研究了无限维双曲空间等距群中有限生成群的凸-紧致表示性质，并通过弯曲技术获得非共轭于特定分类的表面群表示。
Convex cocompact groups with three-dimensional limit sets
本文提出一种构造三维极限集的凸紧致超反射群的方法，通过输入三维 simplicial 复形生成超反射群，并证明其为薄子群。
We obtain upper and lower bounds on the difference between the renormalized volume and the volume of the convex core of a convex cocompact hyperbolic 3-manifold which depend on the injectivity radius of the boundary of the universal cover of the convex core and the Euler characteristic of the boundary of the convex core. These results generalize results of Schlenker obtained in the setting of quasifuchsian hyperbolic 3-manifolds.
Visual maps between coarsely convex spaces
本文将Dydak和Virk关于Gromov双曲空间间诱导边界连续映射的研究，推广至包含多种重要空间类型的粗凸空间之间的映射。
We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling which types of cusp occur. Using these methods we produce the first known example of a 1-cusped hyperbolic 3-manifold that admits a convex projective structure with a type 2 cusp. We also use these techniques to produce new 1-cusped manifolds that admit a convex projective structu…
Coning off totally geodesic boundary components of a hyperbolic manifold
本文研究了双曲流形全测地边界成分锥化后的几何性质，证明了锥化后的空间具有负曲率，并给出了局部凸子集的条件。
Convex co-compact representations of 3-manifold groups
本文证明闭不可约可定向3流形的基本群具有凸余紧表示当且仅当流形是几何的（欧几里得、双曲或欧几里得×双曲）或分解中每部分为双曲，并描述例子结构。
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.
Convex cocompact right-angled Gromov-Thurston polyhedra
针对5维凸余紧双曲反射群的相关问题，本文采用与Lee和Marquis不同的方法，构造了直角情形的对应例子。
The geometric data on the boundary of convex subsets of hyperbolic manifolds
本文证明了具有不可压缩边界的凸余紧双曲流形中，测地凸子集可由无穷远边界上的共形结构与局部凸曲面上的诱导度量或第三基本形式唯一确定，推广了双曲Weyl问题与Ahlfors-Bers定理，并得出拟富克斯流形的相关存在性结果。
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a…
This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differen…
Boundary metric of Epstein-Penner convex hull and discrete conformality
本文研究Epstein-Penner凸包在Minkowski等空间中诱导的边界度量，证明装饰Teichmüller空间与特定度量模空间间存在双射，并给出离散共形性的新3维解释。
Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders
本文证明当相关施赖埃尔图构成双侧扩展图时，凸余紧双曲曲面的覆盖族具有均匀谱隙，并给出新的通用无共振区域，推广了 Brooks、Bourgain-Gamburd-Sarnak 等先前结果。
Convex co-compact groups with one dimensional boundary faces
本文研究射影线性群的凸余紧子群，利用理想边界面维数和粗Hilbert维数，刻画了此类群及朴素凸余紧子群为字双曲或相对双曲的条件。
Volume functions and boundary data of 3-dimensional hyperbolic manifolds
本文综述了凸余紧双曲流形中度量唯一性由边界数据决定及体积函数（如重整化体积）的研究进展，并探讨了与AdS/CFT和Loewner能量的新联系。
Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space
本文研究了双曲三维空间中无限圆图案和凸多面体的特征化问题，证明了当0≤Θ<π时无限圆图案的刚性和统一化定理，并解决了正则圆图案的分类问题。
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