Lokale Analysis auf Fastkomplexen Mannigfaltigkeit

The proposed research is in the field of Several Complex Variables, an area of research in Mathematics with problems whose approaches require geometric, analytic or algebraic techniques, and that interplays with a broad range of - not necessarily mathematical - fields. One of the main thrusts of the proposed research is the study of geometric invariants which are essential tools in the problem of distinguishing one mathematical object from another. A large part of this proposal is devoted to study the links between different geometric and analytic structures, which could lead to a new understanding of some mathematical concepts. For instance, due to the non- existence of local invariants in symplectic geometry, the local analysis on almost complex manifolds recently became one of the most powerful tools in symplectic geometry, making its systematic development relevant. The proposed research has four parts. (a) The first part concerns the Kobayashi metric in almost complex manifolds. This invariant metric is an important tool for the study of function spaces in Several Complex Variables. I would like to study its boundary behavior in order to understand the geometry of domains with curvature. (b) The second part is about hyperbolicity in complex spaces. Many new phenomena in almost complex manifolds occur which distinguish them from complex ones; for instance, D. McDuff constructed a domain with a disconnected pseudoconvex boundary. Jointly with H. Gaussier, I am studying the existence of large pseudoholomorphic curves in this domain. (c) The third part concerns boundary value problems and involves Partial Differential Equations and the regularity of their solutions. (d) Finally, the last part is about the finite jet determination. It was addressed to L. Blanc-Centi and me by B. Lamel whether one can find an alternative insight into this problem by considering a family of invariant discs.