On Belinskii conformality in countable sets of points

The local behavior of plane quasiconformal mappings is investigated. In particular, generalizing the well-known Reich-Walczak problem, we study the possibility for a quasiconformal mapping to be conformal in the sense of Belinskii at a prescribed point or in a prescribed set of points when the modulus of the complex dilatation is a fixed measurable function. The notion of the Belinskii conformality is related to the conception of asymptotical rotations by Brakalova and Jenkins.

     

On the uniqueness problem of harmonic quasiconformal mappings

In this paper, we give an affirmative answer to Sheretov's problem on the uniqueness of harmonic mappings and improve the unique minimal mapping theorem of Reich and Strebel. Meanwhile, we also solve a problem posed by Reich and obtain the uniqueness theorem on related weight functions.

     

On asymptotically nonexpansive mappings in hyperconvex metric spaces

Since bounded hyperconvex metric spaces have the fixed point property for nonexpansive mappings, it is natural to extend such a powerful result to asymptotically nonexpansive mappings. Our main result states that the approximate fixed point property holds in this case. The proof is based on the use, for the first time, of the ultrapower of a metric space.

     

A note on harmonic quasiconformal mappings

In this note we show that a harmonic quasiconformal mapping f=u+iv with respect to the Poincaré metric of the upper half plane onto itself such that v(x,y)=v(y) or u(x,y)=u(x) is a conformal mapping.     

Analytic sets and the boundary regularity of CR mappings

It is shown that if a continuous CR mapping between smooth real analytic hypersurfaces of finite type in extends as an analytic set, then it extends as a holomorphic mapping.

     

Convergence and finite determination of formal CR mappings

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

     

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

     

A counterexample to the Bartle-Graves selection theorem for multilinear maps

We present an example showing that the multilinear version of the Bartle-Graves Selection Theorem is false, even on finite dimensional spaces.

     

Fixed point theorems under the interior condition

We show a fixed point theorem for condensing mappings under a new condition of the Leray-Schauder type. We call it the Interior Condition. We also discuss examples that demonstrate the independence of these two conditions.

     

Isomorphically Expansive Mappings in

We show that for any renorming of , the well known fixed point free mappings by Kakutani, Baillon and others are not nonexpansive.