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Register a new userCondenser capacity and hyperbolic diameter
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Given a compact connected set $E$ in the unit disk $mathbb{B}^2$, we give a new upper bound for the conformal capacity of the condenser $(mathbb{B}^2, E),$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$ we construct a set of diameter $t$ and show by numerical computation that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is of constant hyperbolic width equal to $t$, the so called hyperbolic Reuleaux triangle.
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We apply domain functionals to study the conformal capacity of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.
For compact subsets $E$ of the unit disk $ mathbb{D}$ we study the capacity of the condenser ${rm cap}( mathbb{D},E)$ by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the equilateral triangle has the least capacity.
We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal mappings from the unit disk onto itself and obtain sharp results in this case, too.
We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, $infty$) condition. We show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian component, and the Laplacian has eigenvalues that are close to any integers, with dimension-free quantitative bounds. Under the additional assumption that the space admits a needle disintegration, we show that the spectral gap is almost maximal iff the observable diameter is almost maximal, again with quantitative dimension-free bounds.
We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.
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