arXiv research

Characterizes monodromies of projective structures on finite-type surfaces.

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…

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More gene…

The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.

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…

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 sa…

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 projecti…

The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved t…

Study on projective structures and their foliations on surfaces.

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the ch…

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…

Study on non-classical generating sets in Fuchsian Schottky groups.

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric s…

The paper generalizes deformation results for Fuchsian representations and shows proper affine actions.

Grafting is a surgery on Riemann surfaces introduced by Thurston which connects hyperbolic geometry and the theory of projective structures on surfaces. We will discuss the space of projective structures in terms of the Thurston's geometric parametrization given by grafting. From this approach we will prove that on any…

A new metric model for quasi-Fuchsian space defined by Bers metrics.

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-F…

We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective struc…

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 funct…

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…

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…

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 …

The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M de…

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,…

The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\C$-Fuchsian curves in complex hyperbolic spaces.

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 t…

Study $\mathbb{C}$-Fuchsian subgroups of non-arithmetic lattices.

This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

The paper finds representations of surface groups in SO(4,1) with specific curvature properties.

Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections…

The study of limit cones for multi-Fuchsian representations in $(\mathrm{PSL}_2\mathbf{R})^d$.

We show that any grafting ray in Teichmüller space determined by an arational lamination or a multi-curve is (strongly) asymptotic to a Teichmüller geodesic ray. As a consequence the projection of a generic grafting ray to moduli space is dense. We also show that the set of points in Teichmüller space obtained by integ…

For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.

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 $\ma…

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 t…

In this paper, we study the topology of the boundaries of quasi-Fuchsian spaces. We first show for a given convergent sequence of quasi-Fuchsian groups, how we can know the end invariant of the limit group from the information on the behaviour of conformal structures at infinity of the groups. This result gives rise to…

The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.

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…

We compare two relationships between quadratic differentials and measured geodesic laminations on hyperbolic Riemann surfaces (by foliations or complex projective structures). Each yields a homeomorphism $\ML(S) \to Q(X)$ for any conformal structure $X$ on a compact surface $S$. The main result is that these maps are n…

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

This paper characterizes Fuchsian groups acting on the circle with invariant laminations.

Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.

The study classifies weakly almost Fuchsian manifolds and proves geometric properties.

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 …

Surveying Hitchin representations of Fuchsian groups.

Introduces Fock bundles for studying surface group character varieties.

We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. It follows that Mostow rigidity holds for Fuchsian buildings: if a group acts properly and cocompactly on tw…