Projective structures with degenerate holonomy and the Bers density conjecture

射影结构完整和乐的变化及其在双曲3-流形钻孔中的应用

Variation of holonomy for projective structures and an application to drilling hyperbolic 3-manifolds

本文研究闭曲面射影结构形变中测地线复长度的导数，利用其邻域内 Schwarzian 范数建立界，并应用于非柱形双曲3-流形的锥流形形变。

Complex projective structures with Schottky holonomy

A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its ideal boundary is a closed orientable surface S whose genus is equal to the rank of the Schottky group. This boundary surface is equipped with a (complex) projective structure and its holonomy representation is an epimorphism from pi_1(S) to the Schottky group. We will show that an arbitrary projective structure with the same holonomy representation is obtained by (2 pi-)grafting the basic structure described above.

Some Graftings of Complex Projective Structures with Schottky Holonomy

Let $\mathcal{G}^*(S,ρ)$ be the graph whose vertices are marked complex projective structures with holonomy $ρ$ and whose edges are graftings from one vertex to another. If $ρ$ is quasi-Fuchsian, a theorem of Goldman implies that $\mathcal{G}^*(S,ρ)$ is connected. If $ρ(π_1(S))$ is a Schottky group Baba has shown that $\mathcal{G}(S,ρ)$ (the corresponding graph for unmarked structures) is connected. For the case that $ρ(π_1(S))$ is a Schottky group, this paper provides formulae for the composition of graftings in a basic setting. Using these formulae, one can construct an infinite number of (s…

A generalization of the Epstein-Penner construction to projective manifolds

We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with radial ends, provided the holonomy of each cusp has a fixed point.

Branched projective structures with quasi-Fuchsian holonomy

We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for the space M(k,R). It is shown to …

Projective Structures with (Quasi-)Hitchin Holonomy

In this paper we investigate the properties of the real and complex projective structures associated to Hitchin and quasi-Hitchin representations that were originally constructed using Guichard-Wienhard's theory of domains of discontinuity. We determine the topology of the underlying manifolds and we prove that some of these geometric structures are fibered in a special standard way. In order to prove these results, we give two new ways to construct these geometric structures: we construct them using gauge theory, flat bundles and Higgs bundles, and we also give a new geometric way to construc…

Multi(de)grafting quasi-Fuchsian complex projective structures via bubbles

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ can be joined to the corresponding uniformizing structure $σ_ρ$ by a simple sequence of one bubbling and one debubbling, with a stopover in the space of branched complex projective structures.

2 π-grafting and complex projective structures, I

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich, and Marden asked if 2π-graftings produce all projective structures on $S$ with arbitrarily fixed holonomy (Grafting Conjecture). In this paper, we show that the conjecture holds true "locally" in the space $GL$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $GL$ in the Thurston coordinates. In the sequel paper, using this local solution, we prove the conjecture for generic holonomy.

Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\geq 2$, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor, and we establish a partial converse for hyperelliptic structures.

Holonomy map fibers of $\mathbb{C}{\rm P}^1$-structures in moduli space

Let $S$ be a closed oriented surface of genus $g\geq 2$. Fix an arbitrary non-elementary representation $ρ\colonπ_1(S)\to {\rm SL}_2(\mathbb{C})$ and consider all marked (complex) projective structures on $S$ with holonomy $ρ$. We show that their underlying conformal structures are dense in the moduli space of $S$.

通过扭曲Alexander多项式的单穿孔环面丛的射影刚性

Projective Rigidity of Once-Punctured Torus Bundles via the Twisted Alexander Polynomial

本文通过扭曲Alexander多项式验证双曲单穿孔环面丛的射影刚性，并将其与字符簇切空间上的诱导作用及谱序列中的群论作用联系起来。

Slicing, skinning, and grafting

We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.

The classification of ends of properly convex real projective orbifolds II: Properly convex radial ends and totally geodesic ends

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of properly convex ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. For this, we will study the natural conditions on eigenvalues of holonomy r…

$\mathbf{RP}^n \\# \mathbf{RP}^n$ 及若干其他流形不承认实射影结构

$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure

本文给出一个比已有结果更短的证明，表明当 $n\geq3$ 时 $\mathbf{RP}^n\\#\mathbf{RP}^n$ 及若干其他流形不具有实射影结构，并修正了 Benoist 关于无限循环和乐群闭实射影流形分类中的一个小错误。

Exotic projective structures and quasifuchsian spaces II

Let $P(S)$ be the space of projective structures on a closed surface $S$ of genus $g >1$ and let $Q(S)$ be the subset of $P(S)$ of projective structures with quasifuchsian holonomy. It is known that $Q(S)$ consists of infinitely many connected components. In this paper, we will show that the closure of any exotic component of $Q(S)$ is not a topological manifold with boundary and that any two components of $Q(S)$ have intersecting closures.

Monodromies of Projective Structures on Surface of Finite-type

We characterize the monodromies of projective structures with fuchsian-type singularities. Namely, any representation from the fundamental group of a Riemann surface of finite-type in $PSL_2(\mathbb{C})$ can be represented as the holonomy of branched projective structure with fuchsian-type singularities over the cusps. We made a geometrical/topological study of all local conical projective structures whose Schwarzian derivative admits a simple pole at the cusp. Finally, we explore isomonodromic deformations of such projective structures and the problem of minimizing angles.

Symplectic coordinates on the deformation spaces of convex projective structures on 2-orbifolds

Let $\mathcal{O}$ be a closed orientable 2-orbifold of negative Euler characteristic. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form $ω$ on the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective structures on $\mathcal{O}$. We show that the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective structures on $\mathcal{O}$ admits a global Darboux coordinates system with respect to $ω$. To this end, we show that $\mathcal{C}(\mathcal{O})$ can be decomposed into smaller symplectic spaces. In the course of the proof, we also study the deformation spa…

A Schottky decomposition theorem for complex projective structures

Let S be a closed orientable surface of genus at least two, and let C be an arbitrary (complex) projective structure on S. We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each component has an injective developing map and a discrete and faithful holonomy representation. This decomposition implies that every projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. Along the way, we show that there is an admissible loop on (S, C), along which a grafting can be done.

The grafting map of Teichmuller space

Grafting is a method of obtaining new projective structures from a hyperbolic structure, basically by gluing a flat cylinder into a surface along a closed geodesic in the hyperbolic structure, or by limits of that procedure. This induces a map of Teichmuller space to itself. We prove that this map is a homeomorphism by analyzing harmonic maps between pairs of grafted surfaces. As a corollary we obtain bending coordinates for the Bers embedding of Teichmuller space.

Pentagon representations and complex projective structures on closed surfaces

We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb C)$-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to $\mathrm{PSL…

曲面上的分支实射影结构与表示的几何化

Branched real projective structures on surfaces and geometrisation of representations

本文引入并研究紧致曲面上的分支实射影结构，证明基本群到PSL(3,R)的每个表示均可作为其和乐表示，并探讨了分支度与第二Stiefel--Whitney类的关系。

On the cobordisms of Möbius circles

The boundary of a Möbius manifold carries a canonical Möbius structure. This enables one to define the cobordism group of $n$-dimensional (closed) Möbius manifolds. The purpose of this note is to show that the cobordism group of Möbius circles is zero, i.e., every Möbius circle bounds a Möbius surface. We also complete N. Kuiper's classification of projective structures on $S^1$ (we show that there are in fact two series of projective circles with parabolic holonomy, and not one).

A classification of radial or totally geodesic ends of real projective orbifolds I: a survey of results

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study …

Complex projective structures: Lyapunov exponent, degree and harmonic measure

We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov ex…

A classification of radial and totally geodesic ends of properly convex real projective orbifolds

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study the natural condit…

C-gluing construction and slices of quasi-Fuchsian space

Given a pants decomposition $\mathcal{PC} = \{γ_1, \ldots, γ_ξ\}$ on a hyperbolizable surface $Σ$ and a vector $\underline{c} = (c_1, \ldots, c_ξ) \in \mathbb{R}_+^ξ$, we describe a plumbing construction which endows $Σ$ with a complex projective structure for which the associated holonomy representation $ρ$ is quasi-Fuchsian and for which $\ell_ρ(γ_i) = c_i$. When $\underline{c} \to \underline{0} = (0, \ldots, 0)$ this construction limits to Kra's plumbing construction. In addition, when $Σ= Σ_{1,1}$, the holonomy representations of these structures belong to the `linear slice' of quasi-Fuchs…

The oriented graph of multi-graftings in the Fuchsian case

We prove the connectedness and calculate the diameter of the oriented graph of graftings associated to exotic complex projective structures on a compact surface S with a given holonomy representation of Fuchsian type. The oriented graph of graftings is the graph whose vertices are the equivalence classes of marked CP^1-structures on S with a given fixed holonomy, and there is an oriented edge between two structures if the second is obtained from the first by grafting.

On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doubly decorated structures. In addition, we describe a s…

具有正则奇点的 Schwarzian 方程的单值性

Monodromy of Schwarzian equations with regular singularities

本文刻画有限型负 Euler 示性数穿孔曲面上带正则奇点的 Schwarzian 方程的全部单值性表示，并通过构造指定完整群的复仿射结构处理表观奇点，进而推出带锥点球面度量的完整群。

叶状射影结构的叶

Leaves of Foliated Projective Structures

该研究刻画了闭曲面群Hitchin分支中叶状射影结构的叶射影等价的充要条件，并应用该结果否定了Benzécri关于高维正常凸域射影类空间拓扑的经典问题。

The Compactification of the Moduli Space of Convex RP(2) Surfaces, I

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to t…

可微复结构：带符号空间、嫁接与单值性

Meromorphic Projective Structures: Signed Spaces, Grafting and Monodromy

本文研究了可微复结构的带符号空间，证明了Grafting定理的类比，并展示了单值性映射的局部双全纯性质。

On immersions of surfaces into SL(2,C) and geometric consequences

We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a similar version of the classic Gauss-Codazzi equations, and conversely that solutions of Gauss-Codazzi equations are immersion data of some equivariant map. This study has some interesting geometric consequences: 1) it provides a formalism to study immersions of surfaces into SL(2,C) and into the space of geodesics of H^3; 2) it generalizes the classical theory of…

On Thurston's parametrization of $\mathbb{C}{\rm P}^1$-structures

Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm P}^1$-structures. In this note, we summarize Thurston's parametrization of $\mathbb{C}{\rm P}^1$-structures, based on Kamishima-Tan and Kulkani-Pinkall. We, in addition, give independent proofs for the following well-known theorems on $\mathbb{C}{\rm P}^1$-structures by means of pleated surfaces given by the parameterization. (1) Goldman's Theorem on $\math…

Top terms of polynomial traces in Kra's plumbing construction

Let $Σ$ be a surface of negative Euler characteristic together with a pants decomposition $¶$. Kra's plumbing construction endows $Σ$ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across the $i^{th}$ pants curve is defined by a complex parameter $τ_i \in \C$. The associated holonomy representation $ρ: π_1(Σ) \to PSL(2,\C)$ gives a projective structure on $Σ$ which depends holomorphically on the $τ_i$. In particular, the traces of all elemen…

关于带孔曲面的 $SU(2)$ 特征簇的辛方面

On symplectic aspects of $SU(2)$ character varieties for punctured surfaces

本文研究带奇数孔曲面的 $SU(2)$ 平坦联络模空间的辛性质，证明映射类群到辛映射类群的同态在维数大于2时是单射，并给出5点爆破射影平面中拉格朗日球面的完全分类及其可移动性判定。