Regularity of Bergman, Szegö Kernels, CR manifold embedding

Partial Differential Equations (PDE) are used for modeling most natural phenomena. Up until the 1950s the conventional wisdom was that, under appropriate assumptions, every PDE can be solved. However, Hans Lewy produced an example of an aberrant PDE, which does not possess a solution (event though it should, because it is a very nice and simple PDE). This surprising example started the investigation of solvability of PDEs and the relationship to their regularity theory. Our research project deals with a situation in which the solutions of a large class of PDEs (the tangential CR equations) has a geometric interpretation. In this setting, the PDE corresponds to an abstract geometric object (a CR manifold), and its solvability (integrability) is interpreted as the realizability of this geometric object in a concrete Euclidean space through an embedding. There are local as well global problems in this context, both of them posing their own set of questions. Most of the global solutions deal with CR manifolds which are still rather small, and in any case, strong assumptions on the structure are needed. Few optimal results are known. Our project aims at finding good assumptions for the existence of (local) embeddings on the one hand, and at revealing sufficient conditions for global embeddability on the other hand. In order to do this, a team of researchers from Taiwan and Austria has joined forces to shed new light on this classical problem using methods developed over the last couple of years .

CR geometry studies structures that are closely linked to solutions of a first-order system-the CR differential equations. Pseudoconvexity plays a central role: it provides a robust geometric condition (especially in its strict form) that characterizes natural boundaries of solutions-beyond which they cannot necessarily be continued. Another important tool is integral kernels such as the Szeg kernel (and likewise the Bergman and Leray kernels), which reproduce solutions of the homogeneous CR equations via integral representations. In the project, we investigated, among other things, the usefulness of such kernels for embedding problems, and we highlight one result here as an example. Together with partners in Taiwan, we examined the following question: While "most" strictly pseudoconvex CR structures are not embeddable into spheres-a marked contrast to Riemannian geometry, where manifolds are classically embedded into Euclidean space-we show (with Herrmann and Hsiao) that every strictly pseudoconvex CR structure can be approximated arbitrarily well by structures that do embed into spheres. This result significantly advances the understanding of embeddability in CR geometry. It opens new avenues for analyzing difficult structures through well-understood, sphere-embeddable models. Current questions concern approximations by algebraic structures and precise relationships between a given strictly pseudoconvex CR structure and its sphere-embeddable approximations.