Shapes, fingerprints and rational lemniscates

Conformal models and fingerprints of pseudo-lemniscates

We prove that every function that is meromorphic on the closure of an analytic Jordan domain and sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson and Shapiro by characterizing fingerprints of polynomial pseudo-lemniscates.

Fingerprints, lemniscates and quadratic differentials

We discuss some aspects of the theory of recognition of two-dimensional shapes by means of fingerprints of Jordan curves. An interesting approach to problems on shape recognition suggested by P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and extended further by M.~Younsi reveals the fact that the fingerprints of polynomial lemniscates and, more generally, fingerprints of rational lemniscates can be obtained as solutions to certain functional equations involving Blaschke products. Our main goal here is to develop an approach which relates fingerprints of Jordan curves composed of arcs of trajectori…

On the harmonic measure and capacity of rational lemniscates

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.

Two-dimensional shapes and lemniscates

A shape in the plane is an equivalence class of sufficiently smooth Jordan curves, where two curves are equivalent if one can be obtained from the other by a translation and a scaling. The fingerprint of a shape is an equivalence of orientation preserving diffeomorphisms of the unit circle, where two diffeomorphisms are equivalent if they differ by right composition with an automorphism of the unit disk. The fingerprint is obtained by composing Riemann maps onto the interior and exterior of a representative of a shape in a suitable way. In this paper, we show that there is a one-to-one corresp…

有理双纽线与匹配问题

Rational lemniscates and the matching problem

本文研究复平面若尔当曲线的匹配问题，证明任意1到2之间的豪斯多夫维数都存在有解的若尔当曲线，首次给出有理双纽线之外的解，并利用共形焊接与调和测度给出无解新例及有理双纽线的调和测度刻画。

论有理双纽线的形状

On the Shapes of Rational Lemniscates

该研究证明任何平面欧拉图可被同胚有理双纽线强逼近，推广希尔伯特双纽线定理，还得到龙格定理的精确量化版本及平面连续统被有理映射朱利亚集逼近的新结果。