Quasiconformal mappings and singularity of boundary distortion

We extend a well-known theorem of Jones and Makarov [8] on the singularity of boundary distortion of planar conformal mappings. Using a different technique, we recover the previous result and generalize the result to quasiconformal mappings of the unit ball ⊂_∝ ⁿ , n ≥ 2. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere ∂ and show that this estimate is essentially sharp.     

Quasiconformal and harmonic mappings between Jordan domains

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C ^1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C ^1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C ^1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ^∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).      

Quasiconformal mappings and radii of normal systems of neighborhoods

We establish a new geometric criterion for plane homeomorphisms to belong to the class ofq-quasiconforrnal mappings. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1566–1568, November, 1999.     

Quasiconformal mappings and periodic spectral problems in dimension two

We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem. This work is supported by The Royal Society.     

Quasiconformal extension of biholomorphic mappings in several complex variables

Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC ⁿ. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of . We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of onto itself and give some applications. Partially supported by Grant-in-Aid for Scientific Research (C) no. 14540195 from Japan Society for the Promotion of Science, 2004.     

An egg-yolk principle and exponential integrability for quasiregular mappings

Quasiregular mappings f:ⁿⁿ are a natural generalization of analyticfunctions from complex analysis and provide a theory which isrich with new phenomena. In this paper we extend a well-knownresult of Chang and Marshall on exponential integrability ofanalytic functions in the disk, to the case of quasiregularmappings defined in the unit ball of ⁿ. To this end, an ‘egg-yolk’principle is first established for such maps, which extendsa recent result of the first author. Our work leaves open aninteresting problem regarding n-harmonic functions.     

Homeomorphism with zero Jacobian: Sharp integrability of the derivative

Let N?2. We construct a homeomorphism f∈W1,1(N[0,1],RN) such that Jf=0 almost everywhere and sup0<ε?N−1εN[0,1]_∫|Df|^N−ε<∞. In particular, f∈W1,p(N[0,1],N[0,1]) for all p∈[1,N).     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.