Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbb{C}^n$ with $\mathbb{C}^w$ boundary, Does $f$ extend continously to the boundary? for n=1 and not $\mathbb{C}^w$ boundary I have got the answer from thomas here.

share|improve this question
    
Where is "here"? –  Georges Elencwajg Jun 3 '12 at 16:21
    
    
Ah, thank you, Mex. –  Georges Elencwajg Jun 3 '12 at 17:30
    
:) :) :) :) :) :) :) –  Une Femme Douce Jun 3 '12 at 17:56

1 Answer 1

up vote 2 down vote accepted

It's true if $\Omega$ and $V$ are strictly pseudoconvex with $C^\infty$ boundary. It is also true for real-analytic boundaries if the domains are pseudoconvex.

For the non-pseudoconvex case, I need to check some references, but I'm pretty sure the problem has not been solved in full generality. I'm not aware of any counterexamples (for real analytic boundary.)

Added: You should check out Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math.26, 1-65 (1974)

share|improve this answer
3  
Extension for non-pseudo-convex domains (real analytic boundary) in 2 dimensions: iumj.indiana.edu/IUMJ/… (Diederich and Pinchuk). –  user31373 Jun 3 '12 at 16:53
    
@LeonidKovalev Thanks for the reference, I had a vague recollection that it was known for $n = 2$. –  mrf Jun 3 '12 at 21:39

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.