Math. Z. 186, 1-8 (1984) Mathematische Zeitschrift 9 Springer-Verlag 1984 A Reflection Principle for Proper Holomorphic Mappings of Strongly Pseudoconvex Domains and Applications Joseph A. Cima 1, Steven G. Krantz 2. and Ted J. Suffridge 3 i Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27514, USA z Department of Mathematics, Pennsylvania State University, 215 McAllister Building, University Park, Pennsylvania 16802, USA 3 Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, USA w For n=1,2,.., let B,={z~C": ]zl<l}. Let n<keZ +. Suppose that f: B,~B k is a proper holomorphic mapping. Iff extends to be C 3 on/~, and if n>3, k=n +1, then Webster [12] has shown that, up to composition with automor- phisms of B n and B k, f must be of the form f (Zx,..., z,) = (z i .... , Zn, 0). The same hypothesis when n = 2 implies that, up to composition with automor- phisms of B 2 and B 3, f is one of the four mappings (Z1, Z2)I----*(Z1, Z2,0), (Z1, Z2)I---~ (Zl, Z 1Z 2 , Z2), (Zl,Z2)'-->(Z2,r (Zl, Z2)- " Zl Z2, This result is proved in [5]. The techniques of [12] and [5] involve detailed computation of differential invariants on the boundary. Another paper, philosophically related to the above but involving more elementary techniques, is [1] in which proper holomorphic maps of B n to B, are shown to be automorphisms. More recently this result has been proved on strongly pseudoconvex domains in [2] and in [4]. In the paper [3], which is the starting point for the present work, com- pletely elementary means are used to obtain a reflection principle for suitable maps of B,, to B k. This in turn leads to a proof of the result of [12] assuming only C 2 smoothness to the boundary. Further, detailed information is obtained about the maps even when k > n + 1. The elementary techniques of [3] may be adapted to use on strictly * Supported in part by a grant from the National Science Foundation