Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Abstract

Let f = B 1 /B 2 be a ratio of finite Blaschke products having no critical points having no critical points on ∂D. Then f has finitely many critical level curves (level curves containing critical points of f ) in the disk, and the non-critical level curves of f interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of f , one need only understand the configuration of the finitely many critical level curves of f . In this paper we show that in fact such a function f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if f 1 and f 2 have the same configuration of critical level curves, then there is a conformal map φ such that f 1 ≡ f 2 •φ. We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss-Lucas theorem to rational functions) using level curves.Corollary: 6.2 For every finite Blaschke product B of degree n, there is some n degree polynomial p such that the set G := {z : |p(z)| < 1} is connected, and some conformal map φ : D → G such that B ≡ p • φ on D.

Conformal Models and Fingerprints of Pseudo-lemniscates

“…More precisely, the polynomial f n should have a zero of multiplicity two at the origin and n − 2 more simple zeros located on the negative real axis so that the critical level curve configuration consists of a nested sequence of figure-eight graphs. It was proved in [9] that every possible critical level curve configuration is instantiated by some polynomial, so we may let f n be the polynomial with this configuration, and choose D n similarly as in the case n = 4 above.…”

Conformal Equivalence of Analytic Functions on Compact Sets

“…In this case we say that the functions f 1 and f 2 are conformally equivalent. In the special case that each domain D i is a Jordan domain, and each f i satisfies 1)f i = 0 on ∂D i and 2) |f i | = 1 on ∂D i , the author has given a solution in [5] to this problem in terms of the configurations of the critical level curves of f 1 and f 2 in D 1 and D 2 .…”

“…Theorem 2.4 has at least four fundamentally different proofs [1,3,5,7], and at the end of this section we will briefly discuss each proof.…”

“…This conjecture (which in light of the proof posted on math.stackexchange.com we now refer to as a theorem) may be reformulated as follows. Our main goal in this paper is to employ the geometric level curve methods used in [5] to prove the following analytic and simply connected case of Theorem 1.3. Theorem 3.1.…”

Computing Polynomial Conformal Models for Low-Degree Blaschke Products

“…al. [2] in view of applications to computer vision, which has Theorem 1 as a corollary; • the proof of the first author [6] using critical level curve configurations;…”