Vladimir Gutlyanskii

ON CONFORMAL DILATATION IN SPACE

We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grotzsch-Teichmuller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.

Strong Ring Solutions of Beltrami Equations

Ring homeomorphisms are studied and then applied to an extension of the known Lehto existence theorem for degenerate Beltrami equations. On the basis of this extension, we establish a series of new general integral and measure conditions on the complex coefficient for the existence of ACL homeomorphic solutions. These criteria imply many of the known existence theorems as well as new results.

On the Beltrami Equations with Two Characteristics

The Beltrami equations of the first type

On existence and representation of solutions for general degenerate Beltrami equations

f_{\bar{z}} = \mu(z)f_{z}

The Degenerate Case

(9.1.1) is the basic equation for the theory of quasiconformalmapping in the complex plane; see, e.g., [9,30,44] and [152]. The well-known measurable mapping theorem solves the problem on the existence and uniqueness for the classical case.

Ring Q-Homeomorphisms at Boundary Points

Letf be a quasiconformal mapping with the complex dilation μ. A new condition on μ is introduced for the conformality off at a pointz. The result extends the classical Teichmüller-Wittich-Belinskiî regularity theorem.

BMO- and FMO-Quasiconformal Mappings

We study the general degenerate Beltrami equations in the unit disk in Given an arbitrary analytic function with isolated singularities in we find criteria for the existence of a solution of the form where stands for a regular himeomorphism of onto itself.

The Classical Beltrami Equation ||µ||∞<1

Roughly speaking, this term refers to equation (B) in the case where μ is a measurable complex-valued function in a domain D in ℂ with |μ| 1 a.e. in other parts of D. With further assumptions on μ, every nonconstant solution f : D → ℂ of (B), i.e., an ACL mapping which satisfies (B) a.e., is locally quasiregular (and in particular open, discrete, and sense preserving) in the regions where |μ| 1. These mappings may branch at some isolated points in D or may have folds and cusps. This leads to the notation and study of, what we call, folded quasiregular maps (abbreviated as FQR-maps) and branched folded maps (abbreviated as as BF-maps).