arXiv subjects
Publications and source records attributed to Stamatis Pouliasis.
We study the behavior of the analytic capacity of a compact set under deformations obtained by families of conformal maps depending holomorphically on the complex parameter. We show that, under those deformations, the logarithm of the analytic capacity varies harmonically. We also show that the hypotheses in this result cannot be substantially weakened.
The modulus metric (also called the capacity metric) on a domain $D\subset \mathbb{R}^n$ can be defined as $μ_D(x,y)=\inf\{\mbox{cap}\,(D,γ)\}$, where ${\mbox{cap}}\,(D,γ)$ stands for the capacity of the condenser $(D,γ)$ and the infimum is taken over all continua $γ\subset D$ containing the points $x$ and $y$. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space $(D,μ_D)$.
We study the lemniscates of rational maps. We prove a reflection principle for the harmonic measure of rational lemniscates and we give estimates for their capacity and the capacity of their components. Also, we prove a version of Schwarz's lemma for the capacity of the lemniscates of proper holomorphic functions.
We prove that the Green equilibrium measure and the Green equilibrium energy of a compact set K relative to the domains D and G are the same if and only if D is nearly equal to G, for a wide class of compact sets K. Also, we prove that equality of Green equilibrium measures arises if and only if the one domain is related with a level set of the Green equilibrium potential of K relative to the other domain.