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 nearly the same, differing by a multiplicative factor of -2 and an error term of lower order than the maps themselves (which we bound explicitly).
As an application we show that the Schwarzian derivative of a $\CP^1$ structure with Fuchsian holonomy is close to a $2π$-integral Jenkins-Strebel differential. We also study compactifications of the space of $\CP^1$ structures using the Schwarzian derivative and grafting coordinates; we show that the natural map between these extends to the boundary of each fiber over Teichmuller space, and we describe this extension.
Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore…
To a complex projective structure $Σ$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|ϕ_Σ\|_\infty$ and $\|ϕ_Σ\|_2$ of the quadratic differential $ϕ_Σ$ of $Σ$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination.…
Let $(M,g)$ be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of $M$ related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric $g.$ In particular, if the manifold $(M,g)$ is conformally flat, these derivatives vanish on the conformal group $\Og(p+1,q+1),$ where $\mathrm{dim} (M)=p+q.$ This work is a continuation of [1,4] where the Schwarzian derivative was defined on a manifold endowed with a pro…
The boundary at infinity of a quasifuchsian hyperbolic manifold is equiped with a holomorphic quadratic differential. Its horizontal measured foliation $f$ can be interpreted as the natural analog of the measured bending lamination on the boundary of the convex core. This analogy leads to a number of questions. We provide a variation formula for the renormalized volume in terms of the extremal length $\ext(f)$ of $f$, and an upper bound on $\ext(f)$. We then describe two extensions of the holomorphic quadratic differential at infinity, both valid in higher dimensions. One is in terms of Poinca…
We show that on a Riemann surface lamination locally embedded in $\mathbb{C}^2$, $C^1$ functions (in the sense of the $C^1$ structure of the lamination) are uniform limits of ambient $C^1$ functions, with $L^p$ control on the derivatives along the leaves. This implies that locally in $C^2$, a (1,1) positive closed current dominated by a uniformly laminar current is itself uniformly laminar.
Let $M$ be either a projective manifold $(M,Pi)$ or a pseudo-Riemannian manifold $(M,g).$ We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on $M.$ As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection $Pi$ and the conformal class $[g]$ of the metric, respectively. Furthermore, we compute the first cohomology group of $Vect(M)$ with coefficients into the space of symmetric con…
For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least for the case of virtual orders 2 and 3). As an application, we give univalence criteria for a meromorphic function on the unit disk in terms of the projective Schwarzian derivative of virtual order 3.
This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles and Lyapunov exponents. We deal with various ergodic theorems for such laminations: Random and Operator Ergodic Theorems, (Geometric) Birkhoff Ergodic Theorems, Oseledec Multiplicative Ergodic Theorem and Unique Ergodicity Theorems. Applications of these theorems are also given. In p…
In this article, we show the existence of a nontrivial Riemann surface lamination embedded in $\mathbb{CP}^2$ by using Donaldson's construction of asymptotically holomorphic submanifolds. Further, the lamination we obtain has the property that each leaf is a totally geodesic submanifold of $\mathbb{CP}^2 $ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in $\mathbb{CP}^2$.
Let $S$ be a closed, orientable surface of genus at least 2. The cotangent bundle of the "hyperbolic'' Teichmüller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations, while the cotangent bundle of the "complex'' Teichmüller space can be identified with $\CP$ through the Schwarzian derivative. We prove that the resulting map between the two cotangent spaces, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends.
We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct a family of triangulations of the Teichmüller space of punctures surfaces that interpolates between Penner-Bowditch-Epstein's (using the spine construction) and Harer-Mumford-Thurston's (using Strebel's differentials). Finally, we show (adapting arguments of Dumas) that on a fix…
We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.
Holomorphic realizations of pairs of foliations on Riemann surfaces
University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；Graduate Center of the City University of New York（纽约市立大学研究生中心）；Queens College of the City University of New York（纽约市立大学皇后学院）
Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $ϕ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.
We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of …
A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some marked points). Each point of the map corresponds to a line bundle over the spectral curve so that the map corresponds to a discrete subgroup of the Jacobi variety. We derive an explicit formula for the generic maps using Riemann theta functions, describe the typical singularities a…
We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in ${\rm PSU}(1,\,1)$. As an application, we construct explicitly a new class of hyperbolic metrics with countably many singularities on the unit disc.
We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is the phase space attached to an ordinary differential equation, our space M is the "phase space" attached to a certain elliptic equation, which geometrically describes surfaces of constant Gauss-Kronecker curvature k (k-surfaces), 0<k<1. These hyperbolic properties are : a generic leaf is dense, the union of compact leaves is dense, and stability. They follow from an analogous of the Morse lemma for ge…
