Holomorphic Mappings

  • Conference
  • Final Report (pdf)

    Since the inception of the subject of the function theory of several complex variables, holomorphic mappings have played a pivotal role. In 1906, Henri Poincare proved that there exists no biholomorphic mapping between the polydisc and the unit ball in $n$-dimensional complex space. Thus the two most likely candidates for ``canonical domain'' were shown to be inequivalent. In subsequent years, work of Burns/Shnider/Wells and Greene/Krantz has shown that the biholomorphic inequivalence of domains is a generic phenomenon.

    Yet the notion of biholomorphic equivalence is an important one, since it is the primary device for transferring function theory from one domain to another. Also, the group of biholomorphic self-mappings of a domain can be a useful invariant for classifying domains. One of the central problems of holomorphic mapping theory is to determine whether a biholomorphic mapping will extend smoothly (or real analytically) to the boundary. In one complex variable, this question was settled in the thesis of Painleve. Kellogg, Warschawski, and others developed the result to a highly refined theory. Much less is known in several complex variables.

    The breakthrough result for several complex variables was that of C.\ Fefferman in 1974: a biholomorphic mapping of strongly pseudoconvex domains will extend to a diffeomorphism of the closures of the domains. S. Pinchuk extended the result to proper holomorphic mappings. S. Bell and E. Ligocka found far-reaching techniques that will yield theorems of Fefferman-type on a variety of weakly pseudoconvex domains. In particular, for domains satisfying ``Condition $R$'' (a regularity hypothesis about the Bergman projection), the smoothness-to-the-boundary problem has a positive resolution.

    One of the most natural methods for study the Bergman kernel and Condition R is to use the regularity theory of the $\overline{\partial}$ problem as developed by J. J. Kohn. When the $\overline{\partial}$ problem satisfies a subelliptic estimate, then Condition R is easy to verify. It is now known, thanks to work of David Catlin, that a domain has a subelliptic estimate if and only if the domain is of finite type in the sense of D'Angelo and Kohn. Diederich and Forn\ae ss have proved, for example, that any domain with real analytic boundary is of finite type.

    Thus, using a great deal of machinery from function theory, from partial differential equations, and from geometry, we have been able to show that a reasonably large class of domains will have biholomorphic mappings that extend smoothly to the boundary. Forness and his collaborators, Sibony and his collaborators, Bedford, Bell, Catlin, Pinchuk, Lempert, Barrett, Forstneric, and many others have continued to develop the theory.

    Today there is particular interest in studying biholomorphically invariant metrics (e.g., the Caratheodory, Kobayashi, Bergman, and Sibony metrics). Work of G. M. Henkin and L. Lempert has shown that such metrics have direct applications to the smoothness-to-the-boundary questions discussed above. But invariant metrics also have applications to the study of the boundary behavior of holomorphic functions in Hardy classes (work of Krantz), to the study of domains with non-compact automorphism groups, and to general questions of geometric function theory. The recent books of Kobayashi and of Jarnicki/Pflug bring to closure a chapter of these studies and set a stage for further developments. Even workers in $H^\infty$ control theory (work of Helton) have found use for the formalism of the Kobayashi metric. Holomorphic mappings are fundamental invariant objects in complex function theory. They have applications to partial differential equations, to geometry, to function theory, and to interactions among all these areas.

    This project will bring together a group of top researchers, with proven track records, to study the state of the art in the subject and to launch new research programs in promising areas. We will have a five-day working conference There will be about four lectures each day, but plenty of time will be set aside for interactions in small groups. The conference will be followed by a month-long intensive workshop on holomorphic mappings---with six carefully selected participants. Thus we expect immediate and productive follow-up on the ideas generated at the conference.