A tale of two conformally invariant metrics

Abstract

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.  2005 Elsevier Inc. All rights reserved.

On subharmonic and entire functions of small order: after Kjellberg

“…In [16], and also [5], the relationship between the two metrics when G is multiply connected was investigated and, amongst other results, the following was obtained; see [16,Theorem 6]…”

“…In our main result, which follows, we show that for many functions u ∈ K, corresponding to sets E that are unions of closed intervals, the order and lower order of u can be expressed explicitly in terms of the geometric properties of the set E. We do this by developing a new technique to give upper bounds for the growth of positive harmonic functions defined in multiply connected domains, using a result about the relationship between the Harnack metric and the hyperbolic metric in such a domain [16,5], together with the Beardon-Pommerenke estimate for the density of the hyperbolic metric [6].…”

Multiply connected wandering domains of meromorphic functions: Internal dynamics and connectivity

“…|g 0 (𝑧) + g 1 (𝑧) + g 2 (𝑧) + g 3 (𝑧) − (𝑧 − 𝑚 3 )(log 𝑟 − 𝜖)∕ log 𝑟 − 𝑚 4 | < 𝜖 3 3 , 𝑧 ∈ 𝐸 3 |g 0 (𝑧) + g 1 (𝑧) + g 2 (𝑧) + g 3 (𝑧)| < 𝜖 3 3 , 𝑧 ∈ 𝐿 2 ∪ 𝐿 3 .…”

“…Since the functions get progressively smaller on progressively larger left halfplanes, the sum ∑ 𝑛⩾0 g 𝑛 converges locally uniformly to a transcendental meromorphic function g with infinitely many poles, which are exactly at 𝑚 4𝑘+1 and 𝑚 4𝑘+2 ± 𝑖 for 𝑘 ⩾ 0. Furthermore, for any 𝑧 ∈ 𝐸 𝑛 , g never differs from the model map by more than ∑ 𝑚⩾𝑛 𝜖 3 𝑚 ⩽ 1000𝜖 3 ∕999, which -if 𝜖 > 0 is chosen sufficiently small -is much smaller than 𝜖. If we define the sets Proof.…”

Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity

“…Its relevance to us lies on the fact that, if Ω is a bounded domain, it satisfies χ Ω (z, w) ≤ d Ω (z, w) for all z, w ∈ Ω (see, for instance, [3]). Now, since our positive harmonic function h is f -invariant, we have for any pair z, w ∈ U on different level sets of h the following:…”