Mapping problems in Several Complex Variables

Holomorphic mappings between real submanifolds of complex spaces have special properties. This observation goes back to Poincare at the beginning of the last century, but there remain a lot of questions without satisfactory answers in this field; in this project, I propose to investigate the following problems. It is known that for some classes of real hypersurfaces, biholomorphic mappings which fix a point of the hypersurface and map it into itself are uniquely determined by their derivatives at this point of a predetermined order. I will be interested in two questions which remain open: First, determine the number of derivatives needed from geometric data of the hypersurface; and, second, investigate the role of finite type in the unique determination property. To be of finite type at a given point means (for a real analytic hypersurface) that there exist no complex hypersurfaces contained in the real hypersurface passing through this point. It is known that this condition is not necessary in complex two dimensional space, but all known results on finite determination in higher dimensions assume this finite type condition. The finite determination property described above also holds for mappings between submanifolds of spaces which are not necessarily of the same dimension. However, in this context, a lot less is known than in the equidimensional case. For example, it remains open to prove a bound on the number of derivatives needed which does not depend on the mapping; whether this is naturally so or just because we have not discovered the right theorems yet is an intriguing open question which I intend to pursue. Finite determination also holds for mappings between real submanifolds which are not holomorphic but merely CR. In this context, we have just started making some progress towards understanding when finite determination holds, and I will continue the work in this direction. There are also questions which naturally arise during the investigation of the problems discussed above, related to the regularity of mappings, or their classification, which I will also work on.

Holomorphic mappings between real submanifolds of complex spaces have special properties. This observation goes back to Poincare at the beginning of the last century, but there remain a lot of questions without satisfactory answers in this field; in this project, I propose to investigate the following problems. It is known that for some classes of real hypersurfaces, biholomorphic mappings which fix a point of the hypersurface and map it into itself are uniquely determined by their derivatives at this point of a predetermined order. I will be interested in two questions which remain open: First, determine the number of derivatives needed from geometric data of the hypersurface; and, second, investigate the role of finite type in the unique determination property. To be of finite type at a given point means (for a real analytic hypersurface) that there exist no complex hypersurfaces contained in the real hypersurface passing through this point. It is known that this condition is not necessary in complex two dimensional space, but all known results on finite determination in higher dimensions assume this finite type condition. The finite determination property described above also holds for mappings between submanifolds of spaces which are not necessarily of the same dimension. However, in this context, a lot less is known than in the equidimensional case. For example, it remains open to prove a bound on the number of derivatives needed which does not depend on the mapping; whether this is naturally so or just because we have not discovered the right theorems yet is an intriguing open question which I intend to pursue. Finite determination also holds for mappings between real submanifolds which are not holomorphic but merely CR. In this context, we have just started making some progress towards understanding when finite determination holds, and I will continue the work in this direction. There are also questions which naturally arise during the investigation of the problems discussed above, related to the regularity of mappings, or their classification, which I will also work on.