Fefferman's theorem on extension of biholomorphisms

Remarks

Fefferman's proof studies the geodesics of the Bergman metric, which a biholomorphism maps to geodesics, and uses his expansion of the Bergman kernel to control them near the boundary. Later, Bell and Ligocka gave a shorter proof based on the regularity of the Bergman projection up to the boundary.

Consequence: the restriction of $F$ to $\partial\Omega_1$ is a smooth diffeomorphism onto $\partial\Omega_2$ that preserves the CR structures of the boundaries. So every invariant of the CR structure of the boundary, such as those of the Chern–Moser normal form, is a biholomorphic invariant of the domain. A generic small perturbation of the sphere has a CR structure that is not locally equivalent to that of the sphere. So a generic smooth strongly pseudoconvex domain near the ball is not biholomorphic to the ball.

In one variable the analogous fact is classical: a conformal map between two bounded domains with $C^\infty$ boundary extends smoothly to the closures.

Going deeper

For a student

In several variables, the Riemann mapping theorem fails: by Poincaré's theorem the ball and the polydisc are not biholomorphic, and most domains are not biholomorphic to the ball. Fefferman's theorem turns the classification of strongly pseudoconvex domains up to biholomorphism into the classification of their boundaries up to CR diffeomorphism. The proof via the Bergman projection shows the role of boundary regularity of canonical operators, a theme that extends to many weakly pseudoconvex domains.