### On the Bergman kernel and biholomorphic mappings of pseudoconvex domains

Charles Fefferman
1974 Bulletin of the American Mathematical Society
Communicated by E. M. Stein, October 30, 1973 THEOREM 1. Let D x , D 2^C n be strictly pseudoconvex domains with smooth boundaries and suppose that F\D 1 -^D 2 is biholomorphic (i.e., F is an analytic homeomorphism). Then F extends to a diffeomorphism of the closures, F: D 1 ->D 2 . The main idea in proving Theorem 1 is to study the boundary behavior of geodesies in the Bergman metrics (see [2] ) of Z>i and D 2 . To do so, we use a rather explicit formula for the Bergman kernels of D x and D 2
more » ... We begin with a few definitions. Let D={z e C n |^(z)>0} be a strictly pseudoconvex domain, where ip e C^iC") satisfies grad ^^Oon dD. (1) Let JSf (co) denote the Levi form, i.e. the quadratic form jSf (co) dzdz = ^ V Y i.k dz i otic dZj dz k restricted to the subspace {dz e C n \2 ô (d\pfdz^)\ w dzj=0} of C n . (2) For coi, oe 2 e D, set p(a) x , co 2 ) = |co 1 -co 2 | 2 +|(co 2 -co^ • (dipldco)\ oei \. (See [2] again.) (3) A smooth function y defined on D X D has weight k (where k^.0 is an integer or half-integer) if the following estimate holds. l9>(«>i, (o 2 )\ ^ C(y>(tt>i) + f(co 2 ) + P (co 1 , co 2 )) k (4) Set dtp X(z, oe) = y>(oe) + 2 a 2 4 *doe 4 doe h (z i -co,.) (Zj -(Oi)(zk -Û>*).