John A. Arredondo, Camilo Ramírez Maluendas
In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $Γ<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}/Γ$ is a hyperbolic surface homeomorphic to $S$.
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Fuchsian polyhedra in Lorentzian space-forms
设S是亏格>1的紧致曲面，g是S上常曲率K \in \{-1，0，1\}的度量，且具有负奇异曲率的圆锥奇点.当K=1时，我们增加了一个条件，即可收缩测地线的长度大于2π。证明了在曲率为K的Lorentz空间形式中存在凸多面体曲面P和该空间的等距群G，使得商P/G上的诱导度量等距于（S，g）.此外，对（P，G）是唯一的（直到全球等距）在一个特定类别的凸多面体，即Fuchsian多面体。这扩展了A.D.亚历山德罗夫和Rivin-Hodgson关于球的定理到更高的亏格情况，它也是Labourie-Schlenker定理的多面体版本。
We prove that any metric with curvature $\leq -1$ (in the sense of A. D. Alexandrov) on a closed surface of genus $>1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$. The proof is done by approximation, using a result about isometric immersion of smooth metrics by Labourie--Schlenker.
For a given quasi-Fuchsian representation $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a closed surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $ρ$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the same holonomy.
For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $Σ$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_Σ$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Σ$ and gives the existence of holo…
The Lawson surfaces $ξ_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $Γ_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of $Γ_t$. In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $η_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $η_t$ are shown to depend …
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…
The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\leq s$, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group $Γ_{m,s}$ such that the quotient space $\mathbb{H}/ Γ_{m, s}$ is homeomorphic to $S_{m,s}$ and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and giv…
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。
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
双曲空间中的富克斯多面体是在等距富克斯群作用下不变的多面体曲面（即等距群留下全局不变的全测地曲面，它在其上作用余紧）。凸Fuchsian多面体上的诱导度量等距于亏格大于1的紧致曲面上具有正奇曲率的锥奇点的双曲度量。我们证明，这些指标实际上是实现了一个凸Fuchsian多面体（全球等距）。这扩展了A.D.亚历山德罗夫的一个著名定理。
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 …
Saddle towers with infinitely many ends
证明了亏格为零、有无穷多个端点和一个极限端点（特别是具有无穷大全曲率）的$\mathbb{R}^2\times\mathbb{S}^1$中非周期、适当嵌入的极小曲面的存在性.
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。
Non-periodic Riemann examples with handles
证明了在具有无穷多个平行平面端点的欧几里得空间中，非周期的、完备的、嵌入的极小曲面的单参数族的存在性。特别是，我们能够产生有限属的例子和无限属的准周期的例子。
Spectra of Lorentzian quasi-Fuchsian manifolds
研究洛伦兹准富克斯曼多形体的空间状测地流谱性质，证明其具有离散的鲁厄共振谱，并探讨伪黎曼拉普拉斯算子的解析延拓与量子共振态。
We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and Möller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any $g \leq 4$.
A Note on Riemann Surfaces of Large Systole
我们研究了大收缩问题，这涉及紧凑的双曲黎曼曲面的收缩，最短的noncontractible循环的长度，增长grammically在属。Katz、Schaps和Vishne对Buser和Sarnak的构造进行了推广，它使用了固定余紧算术Fuchsian的主“同余”子群，达到了目前已知的最大增长常数γ= 4/3。我们证明这是最好的可能值γ为这个建设使用算术Fuchsians在同余的情况下。最后一节比较了大收缩问题与类似的大围长问题的正则图。
Genus-One Helicoids from a Variational Point of View
我们用变分方法证明了R^3中存在一个完备的、适当嵌入的亏格为一的极小曲面，它在无穷远处渐近于螺旋面。我们还证明了存在的表面是渐近的螺旋面远离螺旋面的轴，但有无限多的手柄定期安排沿轴。最后，我们证明了这类螺旋面的一些新性质。
Minimal surfaces with positive genus and finite total curvature in $\mathbb{H}^2 \times \mathbb{R}$
我们构造的第一个例子完全，适当嵌入极小曲面在$\mathbb{H}^2 \times \mathbb{R}$有限全曲率和正亏格。这些是由水平悬链或其他非退化被加数的胶合副本构造的。我们还建立了每个水平悬链是非退化的。最后，使用相同的技术，我们能够产生适当的嵌入极小曲面与无限多个结束。每个环形端部具有有限的全曲率并且渐近于垂直的全测地平面。
Affine deformations of a three-holed sphere
每一个具有不可解基本群的完备仿射3-流形M都对应一个非紧双曲曲面S。当Sigma同胚于一个三孔球面时，我们对这种完全仿射结构进行分类。特别地，对于每一个这样的完全双曲曲面σ，变形空间等同于R^3中的两个相反的八分区。此外，每一个M允许一个基本的多面体由弯曲的平面界定。因此M同胚于亏格为2的开实体。作为这一理论的一个具体应用，我们构造了Sp（4，Z）内算术富克斯群的适当仿射变形.
Counting surface subgroups in cusped hyperbolic 3-manifolds
该研究证明了双曲3流形中曲面子群的数量随曲面亏格增长的下界，并构造了具有特定性质的曲面子群示例。
Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group
该研究指出三维欧氏单位球中自由边界极小曲面的拓扑与对称群无法唯一确定曲面，构造出无穷多对非等距、具任意充分大亏格、三个边界分支及对应阶反棱柱对称群的此类曲面对。
Upper bounds on Renormalized Volume for Schottky groups
本文利用亏格和可压缩闭曲线族的组合极值长度，给出边界共形于给定 Riemann 曲面的双曲 Schottky 群重整化体积上界，并部分回应 Maldacena 关于 Schottky 与 Fuchsian 流形体积比较的问题。
We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and Dória to show that the systolic length grows logarithmically with respect to the genus.
Bolza quaternion order and asymptotics of systoles along congruence subgroups
我们详细描述了Bolza曲面的算术富克斯群和相应的四元数阶。这种描述使我们能够证明相应的主同余覆盖在收缩期满足界sys（X）> 4/3 log g（X），其中g是亏格。我们还展示了Bolza群作为一个同余子群，并计算出几个例子的“Bolza双胞胎”（使用岩浆）。像赫维茨三胞胎，这些对应于因式分解的某些合理的素数在环的整数不变迹领域的表面。我们利用随机采样结合的Reidemeister-Schreier算法实现岩浆生成这些表面。
Non-holomorphic surface bundles and Lefschetz fibrations
我们证明了某些稳定化如何产生无穷多个闭的定向4-流形，这些流形是亏格g曲面丛的全空间。Lefschetz纤维化）在亏格h曲面上，并且具有非零特征，但不允许具有任何一个方向的复结构，对于“大多数”（分别为所有）可能的值，g至少为3，h至少为2（分别地，g至少为2，h非负）。
Spectrum of the Laplacian on the Fricke-Macbeath surface
本文研究亏格7的唯一Hurwitz曲面Fricke-Macbeath曲面上的拉普拉斯算子谱，证明其第一特征值有七重性且落在[1.23,1.26]区间，数值确定了对应自同构群的7维表示，还给出了相关Fuchsian群以0为中心的Dirichlet域顶点的代数坐标。
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