The Cauchy-Riemann equations and differential geometry

R. O. Wells Jr.
1982 Bulletin of the American Mathematical Society  
Introduction. In 1907 Poincaré wrote a seminal paper [35] on various topics in several complex variables. In this paper we shall discuss Poincaré's paper and its influence and relationship to certain directions of research in complex analysis and geometry over the past 70 years. Poincaré's paper discusses in some detail real 3-dimensional hypersurfaces in C 2 and the relationship of the geometry of these hypersurfaces to the behavior of holomorphic functions and mappings defined near these
more » ... surfaces. The major topics treated by Poincaré include: (a) Hartogs' phenomenon, discovered only one year earlier by Hartogs [24]; (b) the Riemann mapping problem for domains in C 2 ; (c) an analogous equivalence problem for smooth boundaries of such domains now known as the Poincaré equivalence problem; (d) automorphisms or self-equivalences of domains and boundaries of domains (as in (c) and (d)); and (e) the tangential Cauchy-Riemann equations for a real hypersurface in C 2 . The paper of Hartogs [24] and the paper of E. E. Levi [29] stimulated the major developments in several complex variables in this century. The general theory of several complex variables was summarized first by Osgood [36] who treated the early developments. In 1932 Behnke and Thullen wrote their famous monograph [4] which summarized the state of the art and asked the major questions which occupied researchers for the next 40 years. This culminated in the general theory of Stein manifolds in the 1950's (described, for instance, in Gunning and Rossi [23], Grauert and Remmert [21]) and the later significant interaction of several complex variables with modern partial differential equations (as described by Hörmander [26]). Poincaré's work on Hartogs' theorem relates to the general theory of several complex variables mentioned above. His question about the equivalence of domains was taken up by Bergman and Carathéodory who developed their invariant metrics, both very useful in modern developments (see [3 and 11]). His question about the equivalence of boundaries stimulated significant work by Segre and Cartan in 1931-32 which we discuss in more detail later. On the whole, however, the problems proposed by Poincaré have remained dormant for some time while the general theory of several complex variables was being developed. In some sense he was asking more difficult questions whose solutions required invariants of a considerably higher order
doi:10.1090/s0273-0979-1982-14976-x fatcat:5bbn3aze2vhwrgxixdvuxwakmy