Now 48 pages, no figure. v2: new title, various corrections, results extended to include graph singularities ("interacting particles"). v3: various corrections/improvements, in particular thanks to comments by an anonymous referee
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.
In this paper, we prove that if a quasi-Fuchsian 3-manifold $M$ contains a simple closed geodesic with complex length $\Lscr=l+iθ$ such that $θ/l\gg{}1$, then it contains at least two minimal surfaces which are incompressible in $M$.
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…
In this paper, we prove that if a quasi-Fuchsian 3-manifold contains a minimal surface whose principle curvature is less than 1, then it admits a foliation such that each leaf is a surface of constant mean curvature. The key method that we use here is volume preserving mean curvature flow.
It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists some simple close Jordan curve on $S^2_\infty$ such that there are at least $2^N$ disk-type complete minimal surface in $\mathbb{H}^3$ sharing this Jordan curve as the asymptotic boundary.
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…
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 $M$.
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.
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.
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.
An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichmüller space is represented as a path $γ$, we show that $γ$ joins the conformal structures of the two components of the conformal boundary of $M$. Moreover, we obtain an upper bound for the Teichmüller distance between any two points on $γ$, in particular, the Teichmüller distance between the two components of the conformal boundary of $M$, in term…
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.
We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuchsian.
Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed incompressible surface with mean curvature $H$ . A direct application is the existence of embedded surfaces of prescribed constant mean curvatures with constants in $(-2,2)$.
We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the classical Liouville action on the quasi-Fuchsian deformation space. We prove an equality expressing the holography principle, which relates the Liouville action and the renormalized volume for quasi-Fuchsian groups with parabolic and elliptic elements. We also construct the potentia…
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 …
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 structure at infinity. As an application, we establish a conjecture of Labourie from 1992 regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces.
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.
We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic representation.
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…
This paper gives a new proof of a result of Geoff Mess that the linear holonomy group of a complete flat Lorentz 3-manifold cannot be cocompact in SO(2,1). The proof uses a signed marked Lorentzian length-spectrum invariant developed by G.Margulis, reinterpreted in terms of deformations of hyperbolic surfaces.
Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions of Gamma in SO(1,n) and of SO(1,n) in SO(2,n). We prove that if a representation is Anosov in the sense of Labourie then it is also quasi-Fuchsian. We also show that Fuchsian representations are Anosov : the fact that all quasi-Fuchsian representations are Anosov will be proved in a second part by T…
We show a few propositions in favour of relations between the phase space of 3D gravity, moduli of quasi-Fuchsian groups, global solutions of cosh-Gordon equations and minimal surfaces in hyperbolic spaces.
In a recent paper, Q. Mérigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. It also includes: -- A construction of Dirichlet domains in the context of anti-de Sitter geometry, -- A proof that spatially compact globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit locally CAT(-1) Cauchy hypersurfaces.
Quasi-Fuchsian flows and the coupled vortex equations
Institut für Mathematik, Universität Zürich（苏黎世大学数学研究所）；CNRS & LAMA, Université Paris-Est Créteil（法国国家科学研究中心与拉马实验室，巴黎东克雷泰伊大学）；Department of Mathematics, University of Washington（华盛顿大学数学系）
It is known that the almost-Kaehler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out entire connected components in the moduli space. Our construction hinges on an unexpected link between harmonic functions on certain hyperbolic 3-manifolds and self-dual harmonic 2-forms on associated 4-manifolds.
Let $Γ$ be a finitely generated group, and let $\op{Rep}(Γ, \SO(2,n))$ be the moduli space of representations of $Γ$ into $\SO(2,n)$ ($n \geq 2$). An element $ρ: Γ\to \SO(2,n)$ of $\op{Rep}(Γ, \SO(2,n))$ is \textit{quasi-Fuchsian} if it is faithful, discrete, preserves an acausal subset in the conformal boundary $\Ein_n$ of the anti-de Sitter space; and if the associated globally hyperbolic anti-de Sitter space is spatially compact - a particular case is the case of \textit{Fuchsian representations}, i.e. composition of a faithfull, discrete and cocompact representation $ρ_f: Γ\to \SO(1,n)$ an…
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…
This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there is a convex core. There is also a unique "mean" hypersurface, i.e. with traceless second fundamental form. The mean distance between the mean hypersurface and the lower boundary of the convex core endows the space of affine deformations of a given lattice with an asymmetric norm. …
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 function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We …
For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are two important notions of curvature positivity, the Griffiths positivity and Nakano positivity. We study the consequence of these positivities and the relevant estimates. If $E$ is Griffiths negative over Kähler manifold, then there is a Kähler metric on its total space $E$, and we calculate the curvature and prove the non-positivity of the curvature along the tautological direction. The Nakano positivity can be formulated as a positivity for the Nakano curvature operator and we give estimate the Nakano curvature operat…
We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes spanning that curve. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C^3-smooth curve in the boundary of C^3-…
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…
We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless the representation is discrete and faithful in restriction to an invariant totally geodesic 2-plane of curvature -1. When applied to representations into PSL(2,R) of non-extremal Euler class, our result is a step forward in understanding the space of closed anti-de Sitter 3-manif…
We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isomet…
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 …
A Margulis spacetime is a complete flat Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface S homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when S is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of S. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces. This extends …
We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichmüller theory. In the hyperbolic settings both questions have positive answers for a certain subset of the quasi-Fuchsian manifolds: those containing a closed surface with principal curvatures at most 1. We show that this subset is parameterized by an open domain of the cotangent bundle of Teichmüller space. Thes…
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 character variety, clarifying and generalizing a theorem of S. Kawai. Generalizations of results of C. McMullen are derived, notably quasi-Fuchsian reciprocity. The symplectic geometry is also described in a Hamiltonian setting with the complex Fenchel-Nielsen coordinates on quasi-…
Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings. The proof uses some elementary properties of the hype…
In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles the complex Fenchel-Nielsen coordinates for hyperbolic quasi-Fuchsian manifolds.
This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. …
We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlevé equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [44, 61]. We then express in terms of the Stokes matrices of the 3x3 irregular system the monodromy invariants p_{jk}=Tr(M_jM_k) of the 2-dimensional isomonodromic Fuchsian system with four singularit…
As a generalization of Einstein manifolds, the nearly quasi-Einstein manifolds and pseudo quasi-Einstein manifolds are both interesting and useful in studying the general relativity. In this paper, we study the extended quasi-Einstein manifolds which derive from pseudo quasi-Einstein manifolds. After showing the existence theorem of extended quasi-Einstein manifold, we give some special geometric properties of such manifolds. At the same time, we also discuss the extended quasi-Einstein manifolds with certain soliton like generalised Ricci soliton or Riemann soliton. Furthermore, we construct …
In this paper the notion of quasi-isometry between two Riemannian manifolds has been introduced. This idea is also imposed to study quasi-isometry between two almost contact metric manifolds. Moving further, some curvature properties of two quasi-isometrically embedded almost contact metric manifolds, $N(k)-$contact metric manifolds and Sasakian manifolds are investigated. Next, an illustrative example of a quasi-isometry between two Sasakian structures is constructed. Finally, a relation between the scalar curvature and the quasi-isometric constants for two quasi-isometric Riemannian manifold…
