Biholomorphic Equivalence: Analysis, Algebra and Geometry

Biholomorphic equivalence is one of the fundamental questions of Complex Analysis: When are two objects (e.g. domains or their boundaries) related by a biholomorphic map? If they are, from the point of view of Complex Analysis, they can be regarded as the same. In one complex dimension, the celebrated Riemann Mapping Theorem gives two representatives for simply connected domains in the complex plane, namely, the complex plane and the disc. There is no analogue of the Riemann Mapping Theorem in higher dimensional spaces, and the question of biholomorphic equivalence turns into the classification of the objects under consideration as well as the description of suitable representatives. The research we propose is related to the bihlomorphic equivalence of boundaries of domains. The basic question is, when can two pieces of real-analytic hypersurfaces be mapped onto each other by a biholomorphic map? The description of a representative hypersurface is often referred to as a normal form problem. Normal forms should not only solve the biholomorphic equivalence problem, but also give solutions to analytic questions, such as whether a hypersurface is convexifiable, allows for plurisubharmonic defining functions, and many more. At the present stage, the biholomorphic equivalence problem is only solved in general for the class of Levi-nondegenerate hypersurfaces. Our research centers on degenerate situations, which appear naturally.

The project has decisively moved forward the field of Cauchy-Riemann geometry, in particular the theory of the equivalence problem. We could show that the method of jet parametrization leads to an (in principle) constructive resolution of the equivalence problem for minimal, holomorphically nondegenerate, real-analytic Cauchy-Riemann manifolds. We were also able to show that, surprisingly, the equivalence problems for real-analytic nonminimal Cauchy-Riemann manifolds necessarily differs in the smooth and in the analytic category: There are nonminimal, real-analytic Cauchy- Riemann manifolds which are equivalent by means of a Cauchy-Riemann diffeomorphism, but not by a biholomorphism. This discovery has sparked new interest in the equivalence problem for non- minimal Cauchy-Riemann manifolds, and we have already developed a rather good understanding of what kind of model manifolds one should consider in this setting.The project has also moved forward certain other important aspects of Cauchy-Riemann geometry and Several Complex Variables, and has initiated contacts with other disciplines. In particular, advances were made in the theory of finite determination of smooth Cauchy-Riemann diffeomorphisms, the Levi-problem for certain unbounded domains, and a new proof of the theorem of Faran (unified with the results of Lebl) classifying hyperquadric maps from two-dimensional complex space into three-dimensional complex space was developed.