Quasiconformal Deformations Research Articles

Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of its driving function. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of SLE8/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {SLE}_{8/3}$$\\end{document} loops touching the Jordan curve under quasiconformal deformations.

Two Coefficient Conjectures for Nonvanishing Hardy Functions, I

There are two eminent still open conjectures for nonvanishing holomorphic Hardy functions f(z) on the unit disk. The Hummel–Scheinberg–Zalcman conjecture posed in 1977 extends Krzyz’s conjecture of 1968 to Hp spaces with finite p > 1 and states that Taylor’s coefficients of nonvanishing holomorphic functions f ∈ Hp with norm ‖f‖p ≤ 1 are sharply estimated by |cn| ≤ (2/e)1 − 1/p, with appropriate extremal functions. Both conjectures have been investigated by many authors; however still remain open. The desired Krzyz’s estimate |cn| ≤ 2/e for f ∈ H∞ with ‖f‖∞ ≤ 1 was established only for the initial coefficients cn with n ≤ 5 The only known results for the Hummel–Scheinberg–Zalcman conjecture are that it is true for n = 1 and n = 2 as well as some results for special subclasses of Hp. We prove here that the Hummel–Scheinberg–Zalcman conjecture is true for all spaces H2m with m ∈ N. In the limit as m → ∞, this also provides the proof of Krzyz’s conjecture. Our approach involves deep results from Teichmüller space theory, especially the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces, and special quasiconformal deformations of H2m functions.

Real entropy rigidity under quasi-conformal deformations

We set up a real entropy function h R h_\Bbb {R} on the space M d ′ \mathcal {M}’_d of Möbius conjugacy classes of real rational maps of degree d d by assigning to each class the real entropy of a representative f ∈ R ( z ) f\in \Bbb {R}(z) ; namely, the topological entropy of its restriction f ↾ R ^ f\restriction _{\hat {\Bbb {R}}} to the real circle. We prove a rigidity result stating that h R h_\Bbb {R} is locally constant on the subspace determined by real maps quasi-conformally conjugate to f f . As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree d d real maps of real entropy log ⁡ ( d ) \log (d) . The latter discussion moreover entails a complete classification of maps of maximal real entropy.

Real structures on marked Schottky space

Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity radius bounded away from zero. The space parametrizing quasiconformal deformations of Schottky groups of a fixed rank $g \geq 1$ is the marked Schottky space ${\mathcal M}{\mathcal S}_{g}$; this being a complex manifold of dimension $3(g-1)$ for $g \geq 2$ and being isomorphic to the punctured unit disc for $g=1$. In this paper we provide a complete description of the real structures of ${\mathcal M}{\mathcal S}_{g}$, up to holomorphic automorphisms, together their real part.

Holomorphic motions for unicritical correspondences

We study quasiconformal deformations and mixing properties of hyperbolic sets in the family of holomorphic correspondences zr + c, where r > 1 is rational. Julia sets in this family are projections of Julia sets of holomorphic maps on which are skew-products when r is integer, and solenoids when r is non-integer and c is close to zero. Every hyperbolic Julia set in moves holomorphically. The projection determines a branched holomorphic motion with local (and sometimes global) parameterizations of the plane Julia set by quasiconformal curves.

Group actions, Teichmüller spaces and cobordisms

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism M whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary ∂M. In addition to the standard conformal ergodic action of a uniformhyperbolic lattice on the round sphere S n−1 and its quasiconformal deformations in S n , we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed n-ball into S n ), on non-trivial (n − 1)-knots in S n+1, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from “non-standard” components of its varieties of representations (faithful or with large kernels of defining homomorphisms).

The existence of quasiconformal homeomorphism between planes with countable marked points

We consider quasiconformal deformations of C$\backslash$Z. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to C$\backslash$Z. In particular, we characterize the closed subsets of R whose compliments are quasiconformally equivalent to C$\backslash$Z.

Quasi-conformal deformations of nonlinearizable germs

Let f ( z ) = e 2 π i α z + O ( z 2 ) , α ∈ R f(z) = e^{2\pi i \alpha }z + O(z^2), \alpha \in \mathbb {R} , be a germ of a holomorphic diffeomorphism in C \mathbb {C} . For α \alpha rational and f f of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to f f is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When α \alpha is irrational and f f is nonlinearizable it is not known whether f f admits quasi-conformal deformations. We show that if f f has a sequence of repelling periodic orbits converging to the fixed point, then f f embeds into an infinite-dimensional family of quasi-conformally conjugate germs, no two of which are conformally conjugate.

The Universal Phase Space of AdS3 Gravity

We describe what can be called the "universal" phase space of AdS3 gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the Universal Teichm\"uller space T(1) and is obtained from the correspondence between maximal surfaces in AdS3 and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each T(1) sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS3 spacetime) to the periods of the quadratic differentials arising via the Bers embedding of T(1)xT(1). Our construction also yields a symplectic map from T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial surface setting.

Zygmund functions on the real line and quasiconformal deformations

By a complex approach, we obtain some results concerning Zygmund functions on the real line, which corresponds to some classical results about Zygmund functions on the unit circle.

Multipliers of periodic orbits in spaces of rational maps

AbstractGiven a polynomial or a rational function f we include it in a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier ρ⁄=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function ρ in the space associated to f. The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the space.

Constructing geometrically infinite groups on boundaries of deformation spaces

Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold M = ( H 3 ⋃ Ω G ) / G . Suppose that for each boundary component of M , either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit Γ of quasi-conformal deformations of G such that there is a homeomorphism h from Int M to H 3 / Γ compatible with the natural isomorphism from G to Γ , the given laminations are unrealisable in H 3 / Γ , and the given conformal structures are pushed forward by h to those of H 3 / Γ . Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden’s conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.

A criterion for the failure of Ruelle's property

Ruelle proved that for quasiconformal deformations of cocompact Fuchsian groups, the Hausdorff dimension of the limit set is an analytic function of the deformation. In this paper, we give a criterion for the failure of analyticity for certain infinitely generated groups. In particular, we show that it fails for any infinite abelian cover of a compact surface, answering a question of Astala and Zinsmeister.

Deformation of entire functions with Baker domains

We consider entire transcendental functions $f$ with an invariant (or periodic) Baker domain $U$. First, we classify these domains into three types (hyperbolic, simply parabolic and doubly parabolic) according to the surface they induce when we take the quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmüller space. More precisely, we show that the dimension of this set is infinite if the Baker domain is hyperbolic or simply parabolic, and from this we deduce that the quasiconformal deformation space of $f$ is infinite dimensional. Finally, we prove that the function $f(z)=z+e^{-z}$, which possesses infinitely many invariant Baker domains, is rigid, i.e., any quasiconformal deformation of $f$ is affinely conjugate to $f$.

On quasiconformal deformations of transversely holomorphic foliations

Existence of complex codimension-one transverse structure is studied using the complex dilatation. As an application, a version of quasiconformal surgeries of foliations is considered.

Strong convergence of Kleinian groups

In this paper, we consider algebraically convergent sequences of quasi-conformal deformations of a convex cocompact Kleinian group. We characterize those sequences which converge strongly in terms of their Ahlfors–Bers parameterizations.

Patterson-Sullivan measures and quasi-conformal deformations

In this paper we relate the ergodic action of a Kleinian group on the space of line elements to the conformal action of the group on the sphere at infinity. In particular, we show that for a pair of geometrically isomorphic convex co-compact Kleinian groups, the ratio of the length of the Patterson-Sullivan measure on line element space to the length of its push-forward is bounded below by the ratio of the Hausdorff dimensions of the limit sets. Our primary techniques come from ergodic theory and Patterson-Sullivan theory. 1 Basics and Statement of Results Let Isom+(H) n ≥ 2 be the space of orientation-preserving isometries of Hn. As is well-known, this space of isometries can be given the topology induced by uniform convergence on compact sets. A Kleinian group Γ is a discrete subgroup of Isom+(H). As such, Γ acts discontinuously on Hn, and because we make a standing assumption that the action is torsion-free, the quotient manifoldN = Hn/Γ is a complete Riemannian manifold of constant curvature −1. A Kleinian group Γ also acts as a discrete subgroup of conformal automorphisms of the sphere at infinity Sn−1 ∞ ; this action partitions S n−1 ∞ into two disjoint sets. The regular set ΩΓ is the largest open set in Sn−1 ∞ on which Γ acts properly discontinuously, and the limit set LΓ is its complement. In the case that LΓ contains more than 2 points, the limit set is characterized as being the smallest closed Γ-invariant subset of Sn−1 ∞ . Define the convex hull CH(LΓ) of the limit set LΓ to be the smallest convex subset of Hn so that all geodesics with both limit points in LΓ are contained in CH(LΓ). We can take the quotient of CH(LΓ) by Γ (denoted by C(Γ)); this is

Spinning deformations of rational maps

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as spinning. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges to infinity in the moduli space of rational maps of a fixed degree. When the limit exists, it has either a parabolic fixed point, or a pre-periodic critical point in the Julia set, depending on the combinatorics of the data defining the deformation. The proofs are soft and rely on two ingredients: the construction of a Riemann surface containing the closure of the family, and an analysis of the geometric limits of some simple dynamical systems. An interpretation in terms of Teichmüller theory is presented as well.

Multiple bumping of components of deformationspaces of hyperbolic 3-manifolds

Let M be a hyperbolizable 3-manifold with nonempty incompressible boundary of negative Euler characteristic. Suppose that B 1 ,..., B k is a collection of components of the interior of the space of complete, marked hyperbolic 3-manifolds homotopy equivalent to M , such that for any i, j , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. We prove that there is a geometrically finite hyperbolic structure on int ( M ) which is in the closure of each B i . Moreover, we show that this structure can be constructed so as to admit quasiconformal deformations which also lie in the closure of every B i .

Non-rectifiable limit sets of dimension one

We construct quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. It is known that no divergence type group has such a deformation. The main tools in this construction are (1) a characterization of tangent points in terms of Peter Jones' \beta 's, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3) a construction of quasiconformal deformations of a surface which shrink a given geodesic and whose dilatations satisfy an exponential decay estimate away from the geodesic.