David Kalaj

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.

On harmonic functions and the Schwarz lemma

We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.

On Harmonic Diffeomorphisms of the Unit Disc onto a Convex Domain

We prove a theorem for harmonic diffeomorphisms between the unit disc and a convex Jordan domain, which is a generalization of Heinz theorem [E. Heinz (1959). On one-to-one harmonic mappings. Pacific J . Math ., 9 , 101-105] for harmonic diffeomorphisms of the unit disc onto itself. We give a number of corollaries of the theorem we prove.

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

Let QC(K, g) be a family of K-quasiconformal mappings of the open unit disk onto itself satisfying the PDE Δw = g, g ∈ C(U), w(0) = 0. It is proved that QC(K,g) is a uniformly Lipschitz family. Moreover, if |g| ∞ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as K → 1 and |g| ∞ → 0, so w ∈ QC(K, g) behaves almost like a rotation for sufficiently small K and |g| ∞ .

On the Nitsche conjecture for harmonic mappings in ℝ2 and ℝ3

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsches conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r2+1).Lyzzaiks result states thats<t wheret is the length of the Grötzschs ring domain associated withA(r, 1) (see [5]). Weitsmans result states thats≤1/(1+1/2(r logr)2) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log2r) which improves Weitsmans bound for allr, and Lyzzaiks bound forr close to 1. For three-dimensional space the result states thats≤1/(r−logr).

Radius of close-to-convexity and fully starlikeness of harmonic mappings

Let denote the class of all normalized complex-valued harmonic functions in the unit disk , and let denote the class of univalent and sense-preserving functions in such that . If denotes the harmonic Koebe function whose dilation is , then and it is conjectured that is extremal for the coefficient problem in . If the conjecture were true, then contains the family , where Here, and denote the Maclaurin coefficients of and . We show that the radius of univalence of the family is . We also show that this number is also the radius of the fully starlikeness of . Analogous results are proved for a family which contains the class of harmonic convex functions in . We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.

Harmonic maps between annuli on Riemann surfaces

Let ρΣ = h(|z|2) be a metric in a Riemann surface Σ, where h is a positive real function. Let Hr1 = {w = f(z)} be the family of a univalent ρΣ harmonic mapping of the Euclidean annulus A(r1, 1):= {z: r1 < |z| < 1} onto a proper annulus AΣ of the Riemann surface Σ, which is subject to some geometric restrictions. It is shown that if AΣ is fixed, then sup{r1: ℋr1 ≠ ∅} < 1. This generalizes similar results from the Euclidean case. The cases of Riemann and of hyperbolic harmonic mappings are treated in detail. Using the fact that the Gauss map of a surface with constant mean curvature (CMC) is a Riemann harmonic mapping, an application to the CMC surfaces is given (see Corollary 3.2). In addition, some new examples of hyperbolic and Riemann radial harmonic diffeomorphisms are given, which have inspired some new J. C. C. Nitsche-type conjectures for the class of these mappings.

Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture

Let A andA′ be two circular annuli and let ρ be a radial metric defined in the annulusA′. Consider the class Hρ of ρ−harmonic mappings betweenA andA′. It is proved recently by Iwaniec, Kovalev and Onninen that, if ρ = 1 (i.e. if ρ is Euclidean metric) then Hρ is not empty if and only if there holds the Nitsche condition (and thus is proved the J. C. C. Ni tsche conjecture). In this paper we formulate a condition (which we call ρ−Nitsche conjecture) with corresponds toHρ and defineρ−Nitsche harmonic maps. We determine the extremal mappings with smallest mean distortion for map pings of annuli w.r. to the metricρ. As a corollary, we find that ρ−Nitsche harmonic maps are Dirichlet minimizers among all homeomorphisms h : A → A′. However, outside theρ-Nitsche condition of the modulus of the annuli, within the c lass of homeomorphisms, no such energy minimizers exist. This exte nds some recent results of Astala, Iwaniec and Martin (ARMA, 2010) where it i s considered the caseρ = 1 andρ = 1/|z|.

On Quasi-conformal Harmonic Maps Between Surfaces

It is proved the following theorem, if

Radial extension of a bi-Lipschitz parametrization of a starlike Jordan curve

w