Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.     

Partial sums and the radius problem for some class of conformal mappings

Let $|\frac{{s_n (z)}} {{f(z)}} - 1| $|\frac{{s_n (z)}} {{f(z)}} - 1| when f ∈ $\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^2 - 1} \right| < 1$\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^2 - 1} \right| < 1     

On extremal quasiconformal extensions of conformal mappings

Letf(t, z)=z+tω(1/z) be schlicht for ⋎z⋎>1, ω(z) = Σ _n ^{= 0/∞} a _n z ⁿ ,t>0. The paper considers first-order estimates for the dilatation of extremal quasiconformal extensions off ast→0. This work was initiated during the Special Year in Complex Analysis at the Technion, and was supported in parts by the Samuel Neaman Fund, the Forschungsinstitut für Mathematik, ETH, Zürich, and the National Science Foundation.     

The conformal radius as a function and its gradient image

Let Ω be a domain in with three or more boundary points in andR(w, Ω) the conformal, resp. hyperbolic radius of Ω at the pointw ε Ω/{∞}. We give a unified proof and some generalizations of a number of known theorems that are concerned with the geometry of the surface in the case that the Jacobian of ∇R(w, Ω), the gradient ofR, is nonegative on Ω. We discuss the function ∇R(w, Ω) in some detail, since it plays a central role in our considerations. In particular, we prove that ∇R(w, Ω) is a diffeomorphism of Ω for four different types of domains. This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev.     

On gradient Ricci solitons conformal to a pseudo-Euclidean space

We consider gradient Ricci solitons, conformal to an n-dimensional pseudo-Euclidean space, which are invariant under the action of an (n ? 1)-dimensional translation group. We provide all such solutions in the case of steady gradient Ricci solitons.     

A linear operator associated with the Mittag-Leffer function and related conformal mappings

In the present paper, we introduce a linear operator associated with the Mittag-Leffler function. Some convolution properties of meromorphic functions involving this operator are given.     

On stability of classes of conformal mappings. I

Dedicated to the blessed memory of Ivan Nikolaevich Pesin.     

Variation of the conformal radius

We study the change of the conformal radiusr(U) of a simply connected planar domainU versus the subdomainU _ε consisting of the points of distance at least ε to ∂U. We show that the smallest exponent λ such thatr(U)-r(U _t)=0(e ^λ) satisfies 0.59<λ<0.91. We also show that a well-known conjecture implies . Partially supported by NSF Grant DMS-9970398.     

On Lichtenstein’s theorem about globally conformal mappings

We present a new variational proof of the well-known fact that every Riemannian metric on a two-dimensional, simply connected domain with boundary can be represented by globally conformal parameters. From this the corresponding result for a metric on S² is derived.Received: 1 September 2004, Accepted: 13 September 2004, Published online: 10 December 2004Mathematics Subject Classification: 49Q05, 53A10, 53C42     

On the universal integral means spectrum of conformal mappings near the origin

We improve the local estimate near the origin of the integral means spectrum for conformal mappings obtained in our paper from 2005. We also study some algebraic aspects of higher order forms associated with the given conformal mapping.

     

On the curved wedge condition and the continuity moduli of conformal mappings

A Jordan curveL is studied, which satisfies an interior or exterior wedge condition for wedges of various geometric forms. We obtain estimates of the continuity moduli for the conformal mappings of the exterior (interior) ofL onto the exterior (interior) of the unit disk.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 763–769, June, 1993.