# Introduction

• Vesna Todorčević
Chapter

## Abstract

Geometric Function Theory began as a branch of Complex Analysis dealing with geometric aspects of analytic functions, but has since grown considerably, both in scope and in methodology. It considers, for example, the class of quasiregular mappings proven to be a natural and especially fruitful generalization of analytic functions in the planar case. Another class considered is the class of quasiconformal mappings characterized by the property that there is a constant C ≥ 1 such that infinitesimal spheres are mapped onto infinitesimal ellipsoids in such a manner that the ratio of the longest axis to the shortest axis is bounded from above by C. Injective quasiregular mappings are quasiconformal and conformal mappings in the plane are both harmonic and quasiconformal. Moreover, harmonic mappings are smooth and if they are also quasiregular they are locally quasiconformal in higher dimensions. This gives us a motivation to study harmonic quasiconformal mappings in higher dimensions. Today the study of these classes of mappings is recognized as an important research area of Geometric Function Theory.

Geometric Function Theory began as a branch of Complex Analysis dealing with geometric aspects of analytic functions, but has since grown considerably, both in scope and in methodology. It considers, for example, the class of quasiregular mappings proven to be a natural and especially fruitful generalization of analytic functions in the planar case. Another class considered is the class of quasiconformal mappings characterized by the property1 that there is a constant C ≥ 1 such that infinitesimal spheres are mapped onto infinitesimal ellipsoids in such a manner that the ratio of the longest axis to the shortest axis is bounded from above by C. Injective quasiregular mappings are quasiconformal and conformal mappings in the plane are both harmonic and quasiconformal. Moreover, harmonic mappings are smooth and if they are also quasiregular they are locally quasiconformal in higher dimensions. This gives us a motivation to study harmonic quasiconformal mappings in higher dimensions. Today the study of these classes of mappings is recognized as an important research area of Geometric Function Theory.

In more precise analytic terms, a quasiconformal map f : X → Y  is a homeomorphism of two domains in $$\mathbb {R}^n$$ that is differentiable almost everywhere, such that f belongs to Sobolev space $$W^n_{1,loc}$$ and there is a uniform bound K on the ratio of the largest and smallest absolute value of eigenvalue of a differential of f, valid almost everywhere in X. There are many other alternative definitions of quasiconformal mappings that use, for example, moduli of families of curves or linear dilatation which are more geometric in nature showing that quasiconformality is a fruitful notion. The equivalence of geometric and analytic definitions of quasiconformal mappings has been established for quite some time. The two-dimensional quasiconformal theory was developed by mathematicians including Lars Ahlfors, Lipman Bers, Oswald Teichmüller, Frederick Gehring, and William Thurston in the 20th century. Quasiconformal mappings have compactness properties similar to conformal mappings. For instance, they can be used to form normal families of functions under quite general conditions, which gives them a special place in Geometric Function Theory. The theory of two-dimensional quasiconformal mappings has applications in numerous areas of mathematics such as Teichmüller theory of Riemann surfaces, Complex Dynamics  (the famous No Wandering Domains Theorem of Dennis Sullivan), low dimensional Topology, as well as in Physics (String Theory). Higher dimensional applications have been less versatile, due to the rigidity of quasiconformal mappings, discovered by George Mostow in 1968, .

Harmonic mappings are another natural generalization of conformal mappings and analytic functions. These are used extensively in the study of Teichmüller space. The first significant application was given by M. Wolf . Later R. Schoen  posed a well-known conjecture stating that every quasisymmetric homeomorphism $$u:\partial \mathbb H^2\rightarrow \partial \mathbb H^2$$ admits a unique harmonic quasiconformal extension $$f:\mathbb H^2\rightarrow \mathbb H^2$$. This conjecture was resolved recently by V. Marković in  and his result can be used to find a canonical HQC representative for each class in the Universal Teichmüller space.

Since the theories of harmonic mappings and quasiconformal mappings are both well developed, it is of interest to consider how the corresponding results can be strengthened in the presence of both harmonicity and quasiconformality [1, 74, 75, 85, 102, 117]. While these definitions impose strong limitations, some of the results are unexpected and elegant. One such result is, for example, the preservation of boundary modulus of continuity in the case of the unit ball given in . Harmonic quasiconformal (abbreviated as HQC) mappings in the plane were first introduced by Olli Martio in . Today they are investigated both in the planar and in the multidimensional setting from several different points of view. Unfortunately, the powerful machinery developed for the plane is not available in the space, and so our approach is to combine the analytic and geometric aspects of the theory of quasiconformal mappings together with a number of tools from Harmonic Analysis. Among the topics considered in this book are the boundary behavior, including Hölder and Lipschitz continuity, and the more general moduli of continuity, behavior with respect to natural metrics, especially quasihyperbolic metric, distortion estimates, bi-Lipschitz properties with respect to different metrics, and characterization of boundary mappings. We shall also explain an array of tools used in this study such as the conformal moduli of families of curves, Poisson kernels, estimates from the theory of second order elliptic operators, notions of capacity, subharmonic functions, and Riesz potentials.

In Chap. we introduce basic notions and examples on which the rest of the book relies. Conformal invariants, such as harmonic measure, hyperbolic distance, condenser capacity, and modulus of a curve family are fundamental tools of Geometric Function Theory. The theory of quasiconformal mappings in $$\mathbb {R}^n, n\ge 2\,$$ makes effective use of all these tools. For example, even the definition of quasiconformal mappings can be expressed in terms of moduli of families of curves. (For the planar case n = 2 the reader can see this in the L. Ahlfors book  and in the higher dimensional case in the J. Väisälä book .) The relationship between moduli of continuity of a harmonic quasiregular mapping on the boundary and inside the ball in dimension n = 2 is given in  and in higher dimensions in . Chapter also contains an example of a non-Lipschitz harmonic quasiconformal mapping on the unit ball. In  it is shown that for a wide range of domains, including those with uniformly perfect boundary, Hölder continuity on the boundary implies Hölder continuity (with the same exponent) inside the domain for the class of HQC mappings, a result which does not hold for the class of qc mappings.

In Chap. we introduce some hyperbolic type metrics and explain their connection with the theory of qc mappings. This turns out to be a fruitful theme explored by a number of authors. The geometric properties of domains have been another central theme of research, especially in relation to the boundary phenomena. The geometric nature of the boundary is reflected by the quasihyperbolic metric introduced by Gehring and Palka  as a tool for the study of quasiconformal homogeneity. It turns out that the quasihyperbolic metric is invariant under Euclidean similarities, but it is not invariant under conformal mappings, not even under Möbius transformations. From Gehring and Osgood’s result  it follows that for each domain $$\varOmega \subseteq \mathbb R^n$$ and points x, y ∈ Ω, there exists a quasihyperbolic geodesic and that the quasihyperbolic metric is quasiinvariant under quasiconformal mappings. Another hyperbolic type metric of interest is the distance ratio metric jG. The hyperbolic metric in the unit ball or half space is Möbius invariant. However, the distance ratio metric is not invariant under Möbius transformations. Therefore it is natural to ask what is its Lipschitz constant under conformal mappings or Möbius transformations in higher dimensions. Gehring and Osgood proved that this metric is not changed by more than a factor 2 under Möbius transformations. In Chap. we present a refinement of this result given by Simić and Vuorinen . It turns out that the factor 2 can be improved in the cases of the unit ball and the punctured unit ball in $${\mathbb {R}^n}$$ and that, in fact, the best possible factors can be identified. The Gehring–Osgood theorem provides a Hölder-type estimate for the modulus of continuity of quasiconformal mappings with respect to the quasihyperbolic metrics of the domain and the target domain of the mapping. As shown by Vuorinen in [157, Example 3.10], there is no counterpart of this result for analytic functions in the plane. Recall that the famous Schwarz–Pick lemma provides a Lipschitz-type modulus of continuity estimate for analytic functions of the unit disk into itself with respect to the hyperbolic metrics of the domain and target disks. In recent years there has been a large amount of activity in the study of hyperbolic type metrics [39, 58, 60, 81, 82, 97].

Chapter includes bi-Lipschitz properties of harmonic quasiconformal mappings in the planar and the higher dimensional case. The author has shown in  that HQC mappings between any two proper domains in the plane are bi-Lipschitz with respect to the corresponding quasihyperbolic metrics. In the course of proving this, the author also showed that a sense preserving harmonic mapping between two planar domains has a superharmonic logarithm of the Jacobian. This theorem has found some applications on its own. For instance, Tadeusz Iwaniec  used this result in establishing the minimum principle for the Jacobian determinant, a remarkable novelty which leads us to the new analytic proof of the celebrated Radó–Kneser–Choquet theorem. The result from author’s paper  was also used for higher dimensional generalizations of the Pavlović’s bi-Lipschitz condition. In the joint paper  of K. Astala and the author the bi-Lipschitz property is proved for gradient harmonic quasiconformal mappings in the unit ball $$\mathbb {B}^3$$. However, in higher dimensions, Pavlović’s approach seems difficult to apply. The Lipschitz property follows from the regularity theory of elliptic PDEs established by Kalaj  and by a simple and self-contained argument that works for all dimensions given in . The co-Lipschitz condition is much more difficult to tackle, as it is not even known to hold in higher dimensions when the HQC mappings have nonvanishing Jacobian. Indeed, a famous example by J. C. Wood  shows that Jacobian can vanish for harmonic injective mappings in dimensions higher than two.

In Chap. we present a result of the author from  which solves a problem posed by Pavlović about the functions that are quasi-nearly subharmonic (QNS) in the plane by showing that this class is conformally invariant. An analogous result for regularly oscillating (RO) functions is also proved in the same paper. These results motivated Riihentaus (who introduced the QNS class) and Dovgoshey to partially extend these results to the class of bi-Lipschitz mappings . Since bi-Lipschitz mappings are quasiconformal, the general problem of invariance of QNS and RO classes remained open. In cooperation with P. Koskela, the author solved this problem in  by showing that both classes remain invariant under quasiregular mappings with bounded multiplicity, which includes the quasiconformal case. In the paper  the problem of composition u ∘ ϕ is solved, where u is QNS and ϕ is QC, not only in the plane but also in the space. This generalizes results from , as well as results of Riihentaus and Dovgoshey which were based on .

In Chap. we introduce problems related to characterization of boundary values of harmonic quasiconformal mappings. From many aspects for harmonic quasiconformal mappings, the quantity $$\log J(z,f)$$ seems the natural counterpart of $$\log f'(z)$$. In particular, the question arises if the counterparts of Pommerenke’s and Kellogs’ theorems hold for HQC mappings and $$\log J(z,f)$$ instead of conformal mappings and $$\log f'(z)$$. In the last chapter we also pose some problems in this direction.

## Footnotes

1. 1.

In Chap. , we will formally define the notions of quasiconformal and quasiregular mappings.

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## Authors and Affiliations

• Vesna Todorčević
• 1
• 2