Boundary behaviour of holomorphic functions in several variables

The proposed project belongs to the domain of Complex Analysis in Several Variables. It contains two lines of research, both having to do with the behavior of holomorphic functions at the boundary of their domain of definition. The first topic is an analysis of the algebra of CR functions defined near a point p of the boundary bM of a domain M, in particular regarding the properties of the Taylor development. The known results suggest that some of these properties depend on the existence of a CR peak function at the point p. The second one, instead, is of a global nature: it deals with the study of the Bergman space of certain non-compact complex manifolds M. Since such spaces, in general, can be trivial, we recently introduced a method to construct L^2 holomorphic functions in the case when M admits a large group of holomorphic automorphisms. This method depends, again, on the existence of appropriate holomorphic peak functions.

The majority of the research activities carried out during the project belong to the domain of Complex Analysis in Several Variables, and have mostly revolved around the study of geometric properties of real submanifolds of complex spaces. The main results obtained concern the existence of critical decay rates for germs of CR functions, the construction of suitable invariants for real hypersurfaces of finite type, and the deformation of embeddings between analytic hypersurfaces. Furthermore we considered the problem of characterizing homogeneous submanifolds of Rn, making advances in dimension 2.