Global Holomorphic Extension from Degenerate CR Manifolds: A Unified Approach

This paper presents a unified approach to global holomorphic extension for CR functions defined on degenerate CR manifolds. The study of CR manifolds, which are boundaries of complex domains or more generally real submanifolds of complex spaces, lies at the intersection of complex analysis and differential geometry. While significant progress has been made for non-degenerate CR manifolds, the degenerate case introduces substantial analytical challenges, primarily due to the loss of hypoellipticity for the tangential Cauchy-Riemann operators and the absence of certain pseudoconvexity conditions. Our unified framework integrates techniques from microlocal analysis, $L^2$ methods with weights, and geometric invariant theory to establish novel extension theorems. We define and analyze various notions of degeneracy, including finite type conditions, and demonstrate how these affect the existence and domain of holomorphic extensions. A key aspect of our approach is the development of a refined spectral theory for the CR operator, accounting for the characteristic directions arising from degeneracy. The results provide explicit conditions under which CR functions on such manifolds can be extended globally to a neighboring complex manifold, bridging a critical gap in the theory of several complex variables. The findings have implications for understanding the regularity of solutions to the $bar{partial}_b$ equation and for constructing complex structures near real submanifolds.