The proposed research will study existence and regularity theorems for the natural Neumann boundary problem for the tangential Cauchy-Riemann equations on smoothly bounded domains in CR manifolds. All known existence results for this problem work only on domains with very special defining functions, namely those that depend only on the real and imaginary parts of a single CR-holomorphic function. The key idea of this research is to use the fact that such a defining function provides a codimension-2 foliation (near the boundary but away from characteristic points) by compact CR-submanifolds. By using known estimates for the Kohn Laplacian on the compact leaves, one can reduce the Neumann problem to a (generally non-coercive) elliptic boundary problem in a plane domain. These results are expected to have applications to such problems as the local CR embedding problem, local deformations of CR structures, characterizing domains on which the tangential Cauchy-Riemann complex is solvable, regularity of maps between CR manifolds, and the existence of local frames for CR vector bundles.
Non-technical description:
The geometry of complex manifolds (geometric objects in which complex numbers instead of real numbers can be used as coordinates) has recently begun to play a surprisingly important role in both mathematics and physics. For example, in string theory, physicists postulate that the fundamental particles of matter are actually "quantum strings" that vibrate inside sub-microscopic complex manifolds called Calabi-Yau manifolds. The principal analytic tool for studying complex manifolds is the Cauchy-Riemann equations, a system of partial differential equations that characterizes, among other things, those functions that have complex derivatives. When one studies surfaces within complex manifolds (such as the "branes" that arise in string theory), the Cauchy-Riemann equations need to be replaced by a much more complicated system called the "tangential Cauchy-Riemann equations," which we are just beginning to understand. This proposal will develop new techniques for studying some deep analytic questions surrounding the solvability of the tangential Cauchy-Riemann equations, which are expected to be of fundamental importance in understanding the geometry and analysis of surfaces in complex manifolds.
