Rigidity problems in CR-geometry

Bonnet in the 1860s discovered that two surfaces in Euclidean space, which are isometric and whose second fundamental forms agree, can be mapped by a rigid motion to each other, meaning that there is a Euclidean transformation which maps one surface into the other. More than one hundred years later Webster proved the CR-analogue of Bonnet`s theorem and established a new and still very active field within CR-geometry. One aspect of our project is to address a local version of rigidity. For fixed two manifolds one considers the set of all mappings which send one manifold into the other. On the set of mappings the group of automorphisms of the manifolds induces a group action. We say that a map is locally rigid if in the set of mappings there are only maps close by, which originate from the original map if we compose with automorphisms. It is known that under some restrictive nondegeneracy conditions imposed on the set of maps local rigidity holds true. Our aim is to find new sufficient and necessary conditions for local rigidity. It turns out that properties of the group action of automorphisms on mappings play an important role, which we will also examine within this project. Another topic is the study of the global rigidity problem for mappings of spheres contained in different dimensions. We would like to study and classify sphere mappings in cases, which have not been studied before, where new and interesting phenomena, such as infinite dimensionality of the quotient space of maps with respect to automorphisms, appear.

The project dealt with questions which arise in CR geometry in the field of several complex variables. It treated questions concerning mappings of real submanifolds. The study of such objects has a long research history and is an ongoing, active and international research field. We mainly dealt with the question of local rigidity of mappings. If one has given a mapping between two submanifolds in complex spaces, then the group of automorphisms of the manifolds induces a group action on the space of mappings. A mapping is called locally rigid if nearby the mapping, there are only mappings, which belong to its group orbit. Our results are sufficient conditions and a characterization for local rigidity. The sufficient conditions are provided in terms of linear objects, the infinitesimal deformations, while the characterization is given in terms of non--linear higher order infinitesimal deformations. Infinitesimal deformations are vectors, which are tangent to the target manifold along the image of the mapping. For our conditions we have studied the topological properties of the group action and have used the analytic structure of the space of mappings. Infinitesimal deformations were also studied for sphere mappings between unit spheres in complex spaces of different dimensions. Here we could provide a new characterization of the homogeneous sphere mapping and could give a method to compute infinitesimal deformations of any nondegenerate sphere mapping. For this, the so-called reflection matrix was used, with which we were able to provide simple descriptions of biholomorphic invariants of sphere mappings.