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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

1 vote
Accepted

Growth of preimages of singular values of finite type entire map

At least a variant of your statement is true, see Lemma 3.2 in my paper with Miriam Benini. Let $f$ is a transcendental entire function that is bounded on an unbounded connected set. (This is always t …
Lasse Rempe's user avatar
  • 6,548
4 votes

Equivalence of definitions of quasiconformal surfaces?

EDIT. I am returning to this many years after the fact. The essence of the answer remains unchanged, but I am writing it now as a more detailed proof. Suppose that we are given a surface $S$ with a co …
Lasse Rempe's user avatar
  • 6,548
1 vote
Accepted

Curves in the plane and their number of holes

Here's a slightly more hands-on point of view, using just the Jordan curve theorem (although of course in the end it does come down to the same thing as Euler's formula somehow, as described by Alex.) …
ray's user avatar
  • 687
6 votes

Interesting results for open Riemann surfaces

I hope that the following result by Bishop and myself might be "interesting": Every open Riemann surface is a cover over the sphere, branched over only three points. Equivalently, every open Riemann s …
Lasse Rempe's user avatar
  • 6,548
2 votes

Exponential iterates of a complex number

It depends on what you mean by precise coordinates. I am not sure that I would expect to find a number that has a specific closed form. But then, I do not know how to find a "precise" point where the …
Lasse Rempe's user avatar
  • 6,548
1 vote

Points attracting to 0 are dense in $\mathbb C$

Here is one possible elementary argument (somewhat inspired by my paper with Shen in the Monthly, "The exponential map is chaotic"), which avoids any mention of the classification of Fatou components, …
Lasse Rempe's user avatar
  • 6,548
1 vote

Convergence of analytic covering maps to a covering map

The paper of Detlef Bargmann, "Normal families of covering maps" in Journal d'Analyse, 2001, is relevant to this question; see http://dx.doi.org/10.1007/BF02788084 . His Theorem 1 implies, in particul …
Lasse Rempe's user avatar
  • 6,548
2 votes
Accepted

Looking for a sequence of analytic functions with strange behaviour

Here is an attempt to construct an example. I am going to let $K_1$ and $K_2$ be compact subsets in the sphere $\hat{\mathbb{C}}$, rather than the plane (of course, we can change coordinates to move i …
Lasse Rempe's user avatar
  • 6,548
6 votes

Holomorphic function bounded in a sector with angle $>\pi$

Let $L$ be the boundary of the strip $$ G := \{a + ib\colon a>0, -\pi < b < \pi\}, $$ parameterised in clockwise direction. Define $$ F_0\colon \mathbb{C}\setminus \overline{G}\to\mathbb{C}; \quad z\m …
Lasse Rempe's user avatar
  • 6,548
2 votes

Regularity of a conformal map

I started to write this as comments, but really it is too long for a comment. It is something of an elaboration on Alex's answer. The theory is analogous to the theory of the boundary behaviour of con …
Lasse Rempe's user avatar
  • 6,548
2 votes

Fully invariant measures for rational functions

The measure of maximal entropy is the unique measure that is "fully invariant" in your sense. I believe that this already follows from the original proofs - indeed, it is well-known that if you take a …
Lasse Rempe's user avatar
  • 6,548
2 votes
Accepted

Euclidean length of hyperbolic geodesics for annuli with bounded geometry

Yes, you can say things like this. The easiest way to get these kind of results is probably the Gehring-Hayman theorem. It states that, for two points $z$ and $w$ in a simply-connected domain $D$ or …
Lasse Rempe's user avatar
  • 6,548
2 votes

About a sequence of holomorphic maps from annuli

I am not sure that I understand your question correctly. In particular, I am not sure what $+\infty$ and $-\infty$ mean. As I first understood it, the answer is negative. Let $A(r,1)$ be the annulus …
Lasse Rempe's user avatar
  • 6,548
2 votes
Accepted

Star-shapeness of polynomial tracts containing a single zero

As stated, the answer is negative. I will provide a sketch. Indeed, consider some non-star-like Jordan domain $V$, and a conformal isomorphism $\phi\colon V\to\mathbb{D}$ from $V$ to the unit disc. T …
Lasse Rempe's user avatar
  • 6,548
5 votes
Accepted

A generalization of Liouvilles Theorem for entire functions

To elaborate on my comment, choose three pairwise disjoint Jofrdan $U_1$, $U_2$, $U_3$ whose boundary passes through infinity, and let $\gamma_i\subset U_i$ be a curve to infinity for each $i$. Fix $\ …
Willie Wong's user avatar
  • 38.9k

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