Unique Continuation and Regularity for CR Mapping and Related Partial Differential Equations

Partial differential equations describe dynamic relationships between several mutually dependent quantities. As such, they are ubiquitous in the mathematical modeling of systems in the sciences, engineering, and social sciences. This project will explore the question of unique continuation of maps in several complex variables which, roughly speaking, asks when infinitesimal data about a function that maps one higher dimensional surface to another determines the macroscopic behavior of the function. The project will also support the PI in writing a book for graduate students on recent developments and open questions in function theory. The focus of the award is the theory of CR mappings between CR manifolds and CR functions. Both are fundamental objects in the theory of several complex variables. Specifically, the project will explore four types of problems, namely, unique continuation for CR mappings, regularity of CR mappings and functions, the Rado property for continuous CR functions, and the strong maximum principle for CR functions. Unique continuation will be explored for CR mappings and the closely related second order subelliptic partial differential equations with real analytic coefficients and second order elliptic equations with Gevrey coefficients. The general problem on unique continuation for CR mappings is the following: determine the geometric conditions on two embedded CR manifolds M and M’ so that if f is a CR mapping from M to M’ that vanishes to infinite order at a given point in M, then f vanishes identically in a neighborhood of the point in question. 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.