6

is there a classification theorem for not simply-connected domains in $\mathbb{C}$?

I just know the classification for simply-connected domains, and the classification of annuli.

Do you know the name of this research area, if it is an unsolved problems of complex analysis, let me know.

I would restrict my question to the problem of classification of domains with $\pi_1=\mathbb{Z}$ up to biholomorphism.

If it is easier, you can tell the result for $\mathbb{C}^n$ first.

Evarist
  • 223
  • I'm afraid, I just know that, $\mathbb{C}\backslash{pt}$, $B_1(0)\backslash{pt}$, $\mathbb{C}\backslash [0,1]$ and $B_1(0)\backslash [0,1]$ are not biholomorphic, because of the Riemann-mapping theorem and Riemann removing singularity theorem. – Evarist Jul 22 '13 at 11:28
  • This is partially done in Ahlfors' textbook. – Potato Jul 22 '13 at 14:46
  • I seem to remember (but memory may deceive me) that there is a classification for parallel slit domains (domains - in $\mathbb{C}$ - with finitely many parallel segments removed). – Daniel Fischer Jul 22 '13 at 19:28
  • At first I did not notice that you restricted to $\pi_1=\mathbb Z$ (which is a doubly-connected domain), and answered in more generality. But now I added links specific to doubly connected case. – 40 votes Jul 25 '13 at 17:38

1 Answers1

5

Chapter V of Geometric Theory of Functions of a Complex Variable by Goluzin contains much of what is known about the subject. Among the results obtained after the book was written, most notable is the work of He and Schramm on Koebe's Kreisnormierungsproblem stated below.

Conjecture. For every domain $\Omega\subset \mathbb C$ there is a conformal map of $\Omega$ onto a domain $\Omega'\subset \mathbb C$ all of whose boundary components are circles or points.

  • For simply-connected domains the above is the Riemann mapping theorem.
  • For finitely connected domains it was proved in 1908 by Koebe. (Simpler proof is given in Goluzin's book.)
  • For domains with countably many connected components of the complement it was proved in 1993 by He and Schramm. The map is unique up to a Möbius transformation. (Note: Schramm's Selected Works are in open access, though they do not include everything he did on Kreisnormierungsproblem).
  • In full generality the conjecture remains open. It is known that uniqueness fails for general domains with uncountably many connected components.

Back to classical results, presented in Goluzin's book (with historical notes which I do not reproduce here):

For every doubly-connected domain $\Omega\subset \mathbb C$ there is a conformal map of $\Omega$ onto a circular annulus $\{z: r<|z|<R\}$. The ratio $R/r$ is determined by $\Omega$.

The above was discussed several times here and on MathOverflow: one, two, three.

For every domain $\Omega\subset \mathbb C$ there is a conformal map of $\Omega$ onto a domain $\Omega'\subset \mathbb C$ such that every connected component of $\mathbb C\setminus \Omega'$ is a horizontal line segment or a point.

For finitely connected domains there is a uniqueness statement.

For every domain $\Omega\subset \mathbb C$ and every $\varphi\in [0,\pi/2]$ there is a conformal map of $\Omega$ onto a domain $\Omega'\subset \mathbb C$ such that every connected component of $\mathbb C\setminus \Omega'$ is an arc of logarithmic spiral with pitch $\varphi$.

The special cases $\varphi=0,\pi/2$ correspond to radial slits and arcs of concentric circles. As above, some arcs may degenerate to points. For finitely connected domains there is a uniqueness statement.

Community
  • 1
40 votes
  • 9,736