Invariant analysis in several complex variables, CR and complex geometry.

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The study of invariant geometries is a topic of fundamental importance in mathematics. These geometries arise naturally in many areas, including several complex variables (SCV), partial differential equations (PDE), and algebraic, complex, and differential geometry. This research project by the Principal Investigator is centered around a particular geometry - CR geometry - that arises in the study of SCV and complex geometry. It has deep and profound connections to central topics in mathematical and theoretical physics, including quantum field theory, general relativity, and string theory, as well as applications in systems engineering and control theory. The study of obstruction flatness, e.g., which has a prominent role in this project, has a direct link, via the Lorentzian Fefferman space, to the equations of motion in conformal gravity. The ideas and techniques that are needed for the investigations in this project come from a broad range of mathematical areas: complex analysis/geometry, PDE, and differential geometry; and, at the same time, the techniques and tools developed in this project will benefit these areas as well. The project will also provide interesting research topics and learning experiences for graduate students and postdocs. The seminar activity that will result from the project should be inspiring and stimulating for both students and other researchers. The goal of this mathematics research project is to study geometric and analytic aspects of invariant metrics and their potentials in complex analytic spaces, and related invariants in CR structures. The PI will investigate questions that are motivated by the classical and generalized Cheng-Yau and Ramadanov Conjectures concerning the Bergman metric and Bergman kernel, respectively, in strictly pseudoconvex domains in complex analytic spaces. He intends to work on the Cheng-Yau Conjecture for domains in Stein spaces with isolated singularities. The Ramadanov Conjecture, which asserts that strictly pseudoconvex domains with Bergman log flat boundaries are spherical, has been shown to fail in complex manifolds of dimension at least 3 (but holds true in dimension 2). This opens a compelling classification problem for domains with Bergman log flat boundaries that the PI intends to investigate. He also intends to study the asymptotic boundary behavior of invariant quantities such as the Bergman kernel, Bergman metric, and the Cheng-Yau solution to the complex Monge-Ampere equation. The investigations will elucidate how the boundary geometry influences analysis in complex spaces with boundaries. Additionally, the PI intends to pursue a theory for the Kohn Laplacian in domains on abstract CR manifolds. This is relevant to the local CR embedding problem in dimension 5. He also intends to continue his work on characterizing embeddability of compact CR 3-folds. The PI will investigate consequences of the global vanishing on a compact CR manifold of a higher order local invariant known as the obstruction function. This invariant is the obstruction to smooth extension to the boundary of the Cheng-Yau solution to the complex Monge-Ampere equation. In 3D, this invariant coincides with the trace at the boundary of the log-term in Fefferman's asymptotic expansion of the Bergman kernel and, hence, this problem is also connected with a strong form of the Ramadanov Conjecture. While the PI has made significant progress on the problem in 3D, it is still open in general. In higher dimensions, this problem is separate from the Ramadanov Conjecture, another problem that the PI intends to pursue. Additionally, he will investigate the relationship between the Bergman metric and the complete Kähler-Einstein metric in the setting of Stein spaces with singularities. A particular goal is to settle a generalized Cheng Conjecture in this context. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.