Limits of Holomorphic Extendability from Degenerate CR Boundaries

This paper rigorously investigates the challenging problem of holomorphic extendability of functions and mappings from real submanifolds in complex space, specifically focusing on the behavior near degenerate Cauchy-Riemann (CR) boundaries. While classical theorems, such as Bochner's global holomorphic extension theorem, provide conditions for extendability from smooth, non-degenerate boundaries, these results fundamentally break down when the Levi form degenerates. The degeneracy of the Levi form at a point on a CR boundary signifies a loss of strict pseudoconvexity, introducing complex analytic and geometric subtleties that severely impede the application of standard extension techniques. We explore the nuanced interplay between the degree of degeneracy, the smoothness class of the CR functions, and the local geometry of the boundary, demonstrating that the limits of holomorphic extendability are highly sensitive to these factors. Through a synthesis of contemporary CR geometry, several complex variables theory, and advanced analytical tools, we delineate the conditions under which local or global holomorphic extensions are possible, identifying novel phenomena where extendability can be observed despite degeneracy, albeit often within restricted domains or for specific function classes. Conversely, we also highlight scenarios where even seemingly mild degeneracies impose insurmountable barriers to holomorphic continuation, thereby establishing the critical boundaries for such extensions.