Grunsky inequalities and quasiconformal extension

The Grunsky coefficient inequalities play a crucial role in various problems and are intrinsically connected with the integrable holomorphic quadratic differentials having only zeros of even order. For the functions with quasi-conformal extensions, the Grunsky constant ℵ(f) and the extremal dilatationk(f) are related by ℵ(f)≤k(f). In 1985, Jürgen Moser conjectured that any univalent functionf(z)=z+b ₀+b ₁ z ⁻¹+… on Δ*={|z|>1} can be approximated locally uniformly by functions with ℵ(f)<k(f). In this paper, we prove a theorem confirming Moser’s conjecture, which sheds new light on the features of Grunsky coefficients. In memory of Jürgen Moser The research was supported by the RiP program of the Volkswagen-Stiftung in the Mathematisches Forschungsinstitut Oberwolfach.     

Doubling measures and quasiconformal maps

Part of the research for this paper was done while the author was visiting the Mittag-Leffler Institute. She wishes to thank the Institute for its hospitality. The author was also supported by Grant #DMS 9004251 from the U.S. National Science Foundation.     

Conjugacies between rational maps and extremal quasiconformal maps

We show that two rational maps which are -quasiconformally combinatorially equivalent are -quasiconformally conjugate. We also study the relationship between the boundary dilatation of a combinatorial equivalence and the dilatation of a conjugacy.

     

Bounded turning and quasiconformal maps

We considern-dimensional quasiconformal maps of an arbitrary domain onto a domain with specific properties and prove various results related to the Hayman-Wu theorem.     

Local boundary dilatation of quasiconformal maps

By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary dilatations of points in Teichmuller spaces of simply connected plane domains, which is the same as the usual ones in the case of Jordan domains. Then we show that the answer to the problem of F. Gardiner and N. Lakic is affirmative according to this new definition. Comparing with the usual definitions, the problem of F. Gardiner and N. Lakic is partly answered and the results of G. Cui and Y. Qi is generalized.     

Convergence of characteristics of quasiconformal maps

Ukrainian Mathematical Journal -     

Removability theorems for Sobolev functions and quasiconformal maps

We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces inR ⁿ . The resulting conditions are close to optimal. The first author is supported by N.S.F. Grant No. DMS-9423746. The second author is supported by N.S.F. Grants No. DMS-9304580 and DMS-9706875.     

Evolution of conformal maps with quasiconformal extensions

We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M=DiffS¹/RotS¹ embedded into T. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. This approach allows us to understand the Laplacian growth (Hele-Shaw problem) as a flow in the Teichmüller space.     

Inner estimate and quasiconformal harmonic maps between smooth domains

We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.     

Sets of minimal Hausdorff dimension for quasiconformal maps

For any , there is a compact set of (Hausdorff) dimension whose dimension cannot be lowered by any quasiconformal map . We conjecture that no such set exists in the case . More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.

     

Singular values, quasiconformal maps and the Schottky upper bound

Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ϕdimtortion function ϕ _K (r) and the Agard η-distortion function η _K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.     

Local boundary dilatation of quasiconformal maps in the disk

In this paper we partly give an affirmative answer to a problem proposed by F. Gardiner and N. Lakic by studying the gluing of quasiconformal maps.

     

Extremal problems for quasiconformal maps of punctured plane domains

The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.