1
$\begingroup$

Well I was thinking on this problem, any idea to progress will be gladly accepted, Let $f:\Omega\rightarrow V$ be a biholomorphic map where $\Omega$ and $V$ are bounded open set in $\mathbb{C}$ Does $f$ extend continously to the boundary?

1 Answers 1

1

No, this is in general not true. Standard counterexamples occur on slit annuli $$\{z: 1\le |z|\le R\}\backslash \{z: Im(z) = 0, Re(z) <0\}$$ which cannot be extended periodically along on the circles $\{|z|=r\}$ (like, say, $\log$).

  • 0
    explain me in detail, periodically along the circle means?slit annuli mean?2012-06-03
  • 0
    a slit annulus is a set like the one I defined -- an annulus with a line removed. As for the detail, have a look at the complex $\log$ function, as already suggested in my answer. If you look at $\log$ along the curve $t\mapsto r e^{it}$, with $-\pi < t <\pi$ and $1$t\rightarrow -\pi$ and $t\rightarrow \pi$. – 2012-06-03
  • 0
    If my sets are in $\mathbb{C}^n$ with $\mathbb{C}^w$ boundary then what will be the answer?2012-06-03
  • 0
    @Mex: sorry, can't help you with this one.2012-06-03