Project Details
Abstract
Wider research context
Cauchy-Riemann (CR) geometry is a branch of mathematics where analysis, geometry, and algebra intersect. It uses geometric methods to study the Cauchy-Riemann equations in complex analysis, one of the fundamental partial differential equations, with important applications in physics and engineering. This project studies two fundamental problems in CR geometry:
1. Embeddability: Given an (abstract) CR manifold, under what condition can it be realized as the boundary of some complex manifold?
2. Transversality: When is a CR mapping between CR manifolds of different dimensions CR transversal?
The solution to the local embeddability problem is due to Andreotti and Hill in the real analytic category.
Solutions in the smooth category were given by Boutet, Kuranishi, and Akahori, amongst many others. Past research on the transversality problem focused on equidimensional cases.
Objectives
In this project, we will be interested in studying the following:
1. Spectral behavior of the Kohn Laplacian.
2. Embeddability and the Szegö projection in higher codimension.
3. Heat kernel asymptotics on a complex manifold with boundary.
4. CR transversality in positive codimension cases.
Approach
The two approaches we plan to employ for the embeddability problem are through the spectral theory of the Kohn Laplacian on the one and through the asymptotics of the Szegö kernel on the other hand. Recent advances in these have been made: Spectral stability of the complex Laplacian under deformations of geometric and analytic structures was systematically studied by Fu and myself, and Hsiao and myself established related heat kernel asymptotics for the Kohn Laplacian. Combining these approaches will lead to new insights into the embeddability problem. Transversality is a distinct yet related problem. While almost all results on transversality of CR maps of positive codimension known up to now require stringent conditions on the Levi form of the target. We aim to find optimal geometric conditions that force CR transversality, like in the equidimensional case.
Innovation
We expect our research to be groundbreaking in different ways. The spectral stability of differential operators is a widely studied subject in mathematical physics. Less is known about the spectral behavior of the Kohn Laplacian, which is an archetype of a non-elliptic operator. The parametrix for Szegö kernel in higher codimension has important implications for geometric quantization and dynamical systems.
These problems are not only intrinsically interesting, but they will provide innovative techniques for a wide variety of problems such as the CR embeddability problem. Likewise, our study of transversality in different dimensional cases will lead to the development of new techniques and advances in CR transversality problems.
Primary researchers involved
Principal investigator: Weixia Zhu
Mentor: Bernhard Lamel
Cauchy-Riemann (CR) geometry is a branch of mathematics where analysis, geometry, and algebra intersect. It uses geometric methods to study the Cauchy-Riemann equations in complex analysis, one of the fundamental partial differential equations, with important applications in physics and engineering. This project studies two fundamental problems in CR geometry:
1. Embeddability: Given an (abstract) CR manifold, under what condition can it be realized as the boundary of some complex manifold?
2. Transversality: When is a CR mapping between CR manifolds of different dimensions CR transversal?
The solution to the local embeddability problem is due to Andreotti and Hill in the real analytic category.
Solutions in the smooth category were given by Boutet, Kuranishi, and Akahori, amongst many others. Past research on the transversality problem focused on equidimensional cases.
Objectives
In this project, we will be interested in studying the following:
1. Spectral behavior of the Kohn Laplacian.
2. Embeddability and the Szegö projection in higher codimension.
3. Heat kernel asymptotics on a complex manifold with boundary.
4. CR transversality in positive codimension cases.
Approach
The two approaches we plan to employ for the embeddability problem are through the spectral theory of the Kohn Laplacian on the one and through the asymptotics of the Szegö kernel on the other hand. Recent advances in these have been made: Spectral stability of the complex Laplacian under deformations of geometric and analytic structures was systematically studied by Fu and myself, and Hsiao and myself established related heat kernel asymptotics for the Kohn Laplacian. Combining these approaches will lead to new insights into the embeddability problem. Transversality is a distinct yet related problem. While almost all results on transversality of CR maps of positive codimension known up to now require stringent conditions on the Levi form of the target. We aim to find optimal geometric conditions that force CR transversality, like in the equidimensional case.
Innovation
We expect our research to be groundbreaking in different ways. The spectral stability of differential operators is a widely studied subject in mathematical physics. Less is known about the spectral behavior of the Kohn Laplacian, which is an archetype of a non-elliptic operator. The parametrix for Szegö kernel in higher codimension has important implications for geometric quantization and dynamical systems.
These problems are not only intrinsically interesting, but they will provide innovative techniques for a wide variety of problems such as the CR embeddability problem. Likewise, our study of transversality in different dimensional cases will lead to the development of new techniques and advances in CR transversality problems.
Primary researchers involved
Principal investigator: Weixia Zhu
Mentor: Bernhard Lamel
| Short title | Geom. Analysis auf CR-Mannigfaltigkeiten |
|---|---|
| Status | Active |
| Effective start/end date | 1/05/23 → 30/04/26 |
Keywords
- CR embedding
- CR transversality
- High codimension
- Spectral stability
- Szegö Kernel
- Kohn Laplacian