BLOCH FUNCTIONS, ASYMPTOTIC VARIANCE, AND GEOMETRIC ZERO PACKING

Motivated by a problem in quasiconformal mapping, we introduce a problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity. The problem will be referred to as geometric zero packing, and is somewhat analogous to studying Fekete point configurations. The associated quantity is a density, denoted pc in the planar case, and pH in the case of the hyperbolic plane. We refer to these densities as discrepancy densities for planar and hyperbolic zero packing, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures. The universal asymptotic variance Sigma(2) associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: Sigma(2) = 1- rho H. We obtain the estimates 3.2 x 10(-5) < rho H <= 0.12087, where the upper estimate is derived from the estimate from below on Sigma(2) obtained by Astala, Ivrii, Perala, and Prause, and the estimate from below is more delicate. In particular, it follows that Sigma(2) < 1, which in combination with the work of ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached. Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics B(k, t) similar to 1/4 Sigma(2)vertical bar t vertical bar(2) for small t and k, the conjectured formula B(k, t) = 1/4 k(2)vertical bar t vertical bar(2) is not true. As for the actual numerical values of the discrepancy density rho(C), we obtain the estimate from above rho(C) <= 0.061203 ... by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The value of pH is expected to be somewhat close to that of rho(C).