The Analytic Disc Technique

In this chapter, we present the proof of the CR extension theorem using the technique of analytic discs. The rough idea is the following. In Chapter 13, we showed (without any assumption on the Levi form) that a CR function on a CR submanifold M can be uniformly approximated on an open set w ⊂ M by a sequence of entire functions. To extend a given CR function to an open set Ω in ℂ ⁿ , it is natural to try to show that this approximating sequence of entire functions is uniformly convergent on the compact subsets of Ω. This can be accomplished by the use of analytic discs. Let D be the unit disc in ℂ. An analytic disc is a continuous map A: D → ℂ ⁿ which is holomorphic on D. The boundary of the analytic disc A is by definition the restriction of A to the unit circle S ¹ = ∂D. Often in the literature, the analytic disc and its boundary are identified with their images in ℂ ⁿ . Suppose that {F_j} is a sequence of entire functions that is uniformly convergent to a given CR function f on the open set w ⊂ M. Let us say we wish to show that {F_j } also converges on an open set Ω ⊂ ℂ ⁿ . The idea behind analytic discs is to show that each point in Ω is contained in (the image of) an analytic disc whose boundary image is contained in w. From the maximum principle for analytic functions, the sequence of entire functions {F_j } must also converge uniformly on Ω. So our CR extension theorem is reduced to a theorem about analytic discs, which we state in Section 15.1. In Section 15.2, this analytic disc theorem is established for hypersurfaces. The proof for hypersurfaces involves an easy slicing argument and thus we obtain an easy proof of Hans Lewy’s original CR extension theorem. In Section 15.3, we prove the analytic disc theorem for quadric submanifolds. The proof here is harder than for hypersurfaces but it is still relatively easy since the analytic discs can be explicidy described. The construction of analytic discs for the general case requires the solution of a nonlinear integral equation (Bishop’s equation). This is discussed in Section 15.4. In Section 15.5, we complete the proof of the analytic disc theorem for the general case.