Analysis and Geometry on CR manifolds

This project lies between several complex variables, partial differential equations, and differential geometry. It also has close connections with algebra, operator theory, functional analysis, as well as to contemporary topics in mathematical physics. The main theme of this program is the analysis and geometry on CR manifolds which lays down the framework for many different mathematical problems such as those of sub-Riemann geometry and hypo-elliptic differential operators. Much of the research in this project is motivated by the desire to understand one of the most important objects in this field, namely, the real sub-manifolds in complex space. This program will be on the two important aspects of CR manifolds: the analysis of CR manifolds and CR mappings, and the geometry of strictly pseudo-convex CR manifolds as a counterpart to conformal and Riemannian manifolds. One goal will be to prove the CR transversality of CR mappings between CR manifolds of different dimensions, generalizing recent result by Huang and Zhang to new, and more general, situations. For this purpose, we plan to use an approach in a recent paper by the proposer and Ebenfelt from 2012, as well as some scaling techniques by Huang and Zhang from 2013. Another goal will be to obtain the precise initial regularity of CR mappings that forces them to be smooth or analytic, extending recent result by Lamel and Mir to more general situations. For this purpose, we shall use a reflection principle technique for CR mappings, which has been used successfully in many situations. We intend to refine this technique to apply in new situations. For example, we intend to use this technique to study mappings into general hyper-quadrics in higher dimensional spaces. Moreover, the proposer will also work to bring ideas and techniques from conformal and Riemannian geometries into several complex variables. The proposer will focus on the analysis of Kohn-Laplacian operator and its applications to the geometry of CR manifolds. A recent result on the characterization of CR sphere by eigenvalues of Kohn-Laplacian, an Obata-type theorem in CR geometry, obtained by the proposer, Li, and Wang, encourages us to exploit the spectral geometry of Kohn-Laplacian to study the fine geometry of CR manifolds. Finally, the proposal will also continue his study on Cartan umbilical tensor and umbilical points on strictly pseudo-convex CR manifolds. This study is motivated by the desire to solve an open problem that is similar to the Caratheodory conjecture in differential geometry.

FWF Summaries for Public Relations The project "Geometry and Analysis on CR manifolds" lies between several complex variables, partial differential equations, and differential geometry. It also has close connections with algebra, operator theory, functional analysis, as well as with contemporary topics in mathematical physics. The main theme of this program is the analysis and geometry on CR manifolds which lays down the framework for many different mathematical problems such as those of sub-Riemann geometry and hypo-elliptic differential operators. Much of the research in this project is motivated by the desire to understand one of the most important objects in this field, namely, the real sub-manifolds in complex space. The first result of this project is a study of the geometry of CR submanifolds in a Kaehler manifold with regard to the Tanaka-Webster and Chern connections on the sub- and ambient manifolds, respectively. The project leader found basic equations such as the Gauss-Codazzi equations for the so-called "semi-isometric" CR immersions and demonstrated their usefulness in several applications such as in the estimates of the spectrum of the Kohn Laplacian and the characterization of the "totally umbilical" submanifolds. The project leader and his collaborator Michael Reiter found a concise formula for the well-known Chern-Moser-Weyl tensor for real hypersurfaces given by a general defining function. This is another application of the aforementioned Gauss-Codazzi equations for the semi-isometric CR immersions. Reiter and the project leader have applied the formula to completely resolve a question posed in 2017 by two mathematicians J. Case and R. Gover and provide a solution to the Hirachi's conjecture posed in 2013 in CR geometry. The project leader and his collaborator Bernhard Lamel studied the geometry of CR submanifolds in a CR manifold and constructed a CR analogue of the Ahlfors tensor for CR immersions, generalizing the CR Schwarzian tensor studied earlier by the project leader. Further results along this line were the identifification of the CR Ahlfors tensor and its applications to the study of the spherically equivalent sphere maps. Finally, the project leader and his collaborator Friedrich Haslinger studied the del-complex on the weighted Bergmann spaces on Hermitian manifolds. The del-complex is important for complex analysis and has applications in the representations of the "creation" and "annihilation" operators in quantum mechanics similarly to the duality between the differentiation and multiplication operators in the Segal-Bargman-Fock space. Our study reveals new geometric properties that are important for the duality between the differentiation and multiplication operators in the Segal-Bargmann space---the "flat model"---to hold in the more general setting of the "curved models." It's "boundary counterpart" is the tangential Cauchy-Riemann complex on differential (p,0)-forms with smooth CR coefficients, which should be important for the study of CR manifolds.