Rigidity Problems in CR-Geometry

Basic Information

Funding Source: Austrian Science Fund (FWF)

Duration: 01.05.2016 - 30.06.2019

Principal Investigator: Michael Reiter

Project Number: P28873 Stand-Alone-Project

Description

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.