Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRemarks on the scale invariant Cassinian metric
61
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the geometry of the scale invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp distortion inequalities for the scale invariant Cassinian metric under Mobius transformations.
rate research
Read More
A one-component inner function $Theta$ is an inner function whose level set $$Omega_{Theta}(varepsilon)={zin mathbb{D}:|Theta(z)|<varepsilon}$$ is connected for some $varepsilonin (0,1)$. We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for $0<p<infty$, the derivative of a one-component inner function $Theta$ is a member of the Hardy space $H^p$ if and only if $Theta$ belongs to the Bergman space $A_{p-1}^p$, or equivalently $Thetain A_{p-1}^{2p}$.
We investigate the Plateau and isoperimetric problems associated to Feffermans measure for strongly pseudoconvex real hypersurfaces in $mathbb C^n$ (focusing on the case $n=2$), showing in particular that the isoperimetric problem shares features of both the euclidean isoperimetric problem and the corresponding problem in Blaschkes equiaffine geometry in which the key inequalities are reversed. The problems are invariant under constant-Jacobian biholomorphism, but we also introduce a non-trivial modified isoperimetric quantity invariant under general biholomorphism.
We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.
In this paper, following Grothendieck {it Esquisse dun programme}, which was motivated by Belyis work, we study some properties of surfaces $X$ which are triangulated by (possibly ideal) isometric equilateral triangles of one of the spherical, euclidean or hyperbolic geometries. These surfaces have a natural Riemannian metric with conic singularities. In the euclidean case we analyze the closed geodesics and their lengths. Such surfaces can be given the structure of a Riemann surface which, considered as algebraic curves, are defined over $bar{mathbb{Q}}$ by a theorem of Belyi. They have been studied by many authors of course. Here we define the notion of connected sum of two Belyi functions and give some concrete examples. In the particular case when $X$ is a torus, the triangulation leads to an elliptic curve and we define the notion of a peel obtained from the triangulation (which is a metaphor of an orange peel) and relate this peel with the modulus $tau$ of the elliptic curve. Many fascinating questions arise regarding the modularity of the elliptic curve and the geometric aspects of the Taniyama-Shimura-Weil theory.
In this paper we study the global geometry of the Kobayashi metric on convex sets. We provide new examples of non-Gromov hyperbolic domains in $mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon ewline -vex, bounded and unbounded. Our first aim is to prove that if $Omega$ is a bounded weakly linearly convex domain in $mathbb{C}^n,,ngeq 2,$ and $S$ is an affine complex hyperplane intersecting $Omega,$ then the domain $Omegasetminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $Omegasetminus S,$ where $S$ is relatively closed in $Omega$ and $Omega$ is a convex domain, depends only on that how $S$ looks near the boundary, i.e., whether $S$ near $partialOmega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
