The conformal radius as a function and its gradient image

F. G. Avkhadiev; K. -J. Wirths

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The conformal radius as a function and its gradient image

机译：共形半径的函数及其梯度图

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Let Ω be a domain in $bar {mathbb{C}}$ with three or more boundary points in $bar {mathbb{C}}$ 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 $s_Omega = { (w,h)|w in Omega ,h = R(w,Omega )} $ 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.

机译：设Ω为$ bar {mathbb {C}} $中的一个域，在$ bar {mathbb {C}} $中具有三个或更多边界点，R（w，Ω）为保形的。在点εΩ/ {∞}处的双曲半径Ω。我们给出了一个统一的证明，并对一些已知定理进行了概括，这些定理与表面的几何形状有关$ s_Omega = {（ω，h）| w在Omega中，h = R（w，Omega）} $ ∇R（w，Ω）的雅可比行列式，即R的梯度，对Ω是负的。我们将详细讨论函数∇R（w，Ω），因为它在我们的考虑中起着核心作用。特别是，我们证明∇R（w，Ω）是Ω对四种不同类型域的微分。

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《Israel Journal of Mathematics》 |2005年第1期|349-374|共26页
作者
F. G. Avkhadiev; K. -J. Wirths;
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Chebotarev Research Institute Kazan State University;

Institut für Analysis and Algebra Technische Universität Braunschweig;

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The conformal radius as a function and its gradient image

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