Embedding strictly pseudoconvex domains into balls

Abstract

Every relatively compact strictly pseudocc)nvex domain D with C2 boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each n > 2 there exist bounded strictly pseudoconvex domains D in Cn with real-analytic boundary such that no proper holomorphic map from D into any finite dimensional ball extends smoothly to D. 0. Introduction. In this paper we study the representations of bounded strictly pseudoconvex domains D c cn. If the boundary of D is of class Ck, k E {2, 3, . . ., oo}, then, by a theorem of Fornaess [8] and Khenkin [13], D can be mapped biholomorphically onto the intersection xn Q of a bounded strictly convex domain Q c CN with Ck boundary and a closed complex submanifold X defirXed in a neighborhood of Q in CN, X intersecting the boundary of Q transversally. Moreover, the map f: D X n Q extends to a holomorphic map on a neighborhood of D. The convex domain Q depends on D; hence a natural question is whether a similar result holds with Q replaced by the unit ball B = (z = (Z1L, * , ZN) E CN ||Z||2 = E |Zjl2 n that intersects bBN transversally such that D is biholomorphically equivalent to xn E3ff? This question has been mentioned by Lempert [14], Pinduk [20], Bedford [2] and others. Our main result is that the answer to this question is negative in general. If D is as above, then every biholomorphism of D onto XnE3N extends smoothly to D according to [3]. However, we will show that not all such domains D admit a proper holomorphic map into a finite dimensional ball that is smooth on D (Theorem 1.1). A similar local result was obtained independently by Faran [7]. We shall show that the answer to the question (Q) is positive if we allow the intersections of complex submanifolds with strictly convex domains Q c CN with real-analytic boundaries (Theorem 1.2). ThetheoremsofFornaess [8] andKhenkin [13] onlygiveanQwith smooth boundary. Received by the editors April 4t 1985. 1980 Mathematics Subject Classification. Primary 32H05. 1Research supported by a fellowship from the Alfred P. Sloan Foundation. (t)1986 American Mathematical Society 0002-9947/86 $1 .00 + $ .25 per page