Dynamical Methods in CR-geometry

The subject of Cauchy-Riemann Geometry (shortly: CR-geometry) was initiated in the classical work of Poincare and Cartan, and was further developed in later work of Tanaka, Chern and Moser, and others. CR-geometry is remarkable in that it lies on the border of several mathematical disciplines (Complex Analysis, Differential Geometry and Partial Differential Equations), and is an important tool for each of these areas. In our joint research (partly joint with Rasul Shafikov), we have discovered a new face of CR- geometry. This is a bridge technique between CR-geometry and one of the fundamental areas of modern mathematics: Dynamical Systems. The technique allows for producing a certain vocabulary between the two theories. We call the latter method the CR (Cauchy-Riemann manifolds) - DS (Dynamical Systems) technique. The CR -- DS technique has recently enabled us to solve a number of long-standing problems in CR- geometry, related to important degenerate structures appearing in CR-geometry. The current research project is aimed to develop the CR -- DS technique in several directions, by employing modern methods in Dynamical Systems. We are going to actively collaborate with the Russian team, which consists of well known experts in both CR-geometry and Dynamical Systems. As a bi-product, we plan establishing certain new results in the area of Analytic Differential Equations.

In this research project, on the one hand we studied problems in Cauchy-Riemann (CR) geometry using methods from the theory of dynamical systems, and on the other hand, we also tried to understand dynamical systems from the CR point of view. This is possible because of a bridge between these two very different fields: It is possible to attach to any CR structure a dynamical system. Vice versa, the dynamical systems which appear as attached systems have special properties, and one of the basic questions of interest is the classification of these properties as one crosses bridge. We now summarize two of the main results obtained. One of the basic problems in CR geometry is the classification of real hypersurfaces in complex spaces under the action of so-called biholomorphic maps. These maps are characterized by satisfying a certain partial differential equation, namely the Cauchy-Riemann equations. In several variables the study of the problem dates back to the beginning of the 20th century, when Poincaré observed that there are a huge number of non-equivalent classes under this action. In two complex dimensions, so-called strictly pseudoconvex hypersurfaces were already characterized by Elie Cartan in the 1930s. However, the case of real-analytic hypersurfaces containing a complex hypersurface (which are called nonminimal) remained open. The main difference to the case studied by Cartan is that in his case, when one crosses the bridge into dynamical systems, one turns up with an ordinary differential equation, while in the nonminimal case, this equation becomes singular. The classification of the associated singular equations is considerably harder than in the nonsingular case. We were able to provide a satisfactory solution to this problem in two complex variables. The second result that we would like to discuss is also closely related to the biholomorphic equivalence problem already discussed. In order to formulate it, we have to introduce the notion of a formal equivalence: When one studies the biholomorphic equivalence problem, there is a sequence of purely algebraic obstructions to the existence of an equivalence. When one is able to algebraically solve these equations, one obtains a formal equivalence. The problem is that a formal equivalence is not automatically related to any type of real map. Often, one studies this by trying to ascertain the convergence of the formal equivalence. However, in the nonminimal case, it is known that there are divergent formal equivalences. We were able to construct, using some sophisticated techniques from dynamical systems, that any formal equivalence gives rise to a smooth CR equivalence whose Taylor series coincides with the formal equivalence. Summarizing, we were able to develop methods to successfully cross the bridge between CR geometry and dynamical systems, which were used to prove impactful results.