Holomorphic automorphism of strictly psudo-convex domain smooth on boundary - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:06:28Z http://mathoverflow.net/feeds/question/45666 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45666/holomorphic-automorphism-of-strictly-psudo-convex-domain-smooth-on-boundary Holomorphic automorphism of strictly psudo-convex domain smooth on boundary Craig 2010-11-11T08:34:28Z 2010-11-11T08:34:28Z <p>I am wondering if anything is known about this. I couldn't find anything in the literature.</p> <p>In '74 C. Fefferman published a solution to the following problem. Let $\sigma:D\rightarrow D$ be an automorphism of a strictly pseudoconvex domanain $D\subset\mathbb{C}^n$. Then $\sigma$ extends to a smooth map $\sigma:\overline{D}\rightarrow\overline{D}$.</p> <p>My question: Is anything known about this problem for a strictly pseudoconvex domain $D\subset M$ in a complex manifold $M$?</p> <p>I have an idea of how to prove it for $K_M >0$, i.e. positive canonical bundle. Fefferman's approach used the Bergman metric. The Bergman metric is nondegenerate for more general domains only if $K_M$ is very ample, which is too strong an assumption to be very interesting. </p>