The $\overline \partial$-Neumann problem on strongly pseudo-convex manifolds

Abstract

In this chapter we present a simplification of the recent solution due to J. J. Kohn ([1], [2]) of the so-called $\overline \partial$-Neumann problem introduced by Garabedian and Spencer for complex exterior differential forms on a compact complex-analytic manifold with strongly pseudo-convex boundary. The problem in its present form was investigated by D. C. Spencer and J. J. Kohn by means of integral equations. The present author [13] solved this problem for the special cases of 0-forms and $\bar z$-1-forms (i.e. forms of the types (0,0) and (0,1) in our current notation) on certain “tubular” manifolds and used those results to prove that any compact real-analytic manifold can be analytically embedded in a Euclidean space of sufficiently high dimension. Unfortunately there is an error in that paper which is corrected in. § 8.2 by using the results of Kohn presented in this chapter. These results apply to forms of arbitrary type (p, q) and the solution forms are shown to be of class C^∞ on the closed manifold provided the metric, boundary, and non-homogeneous term ∈C^∞ there. Recently Hörmander [2] has extended these results using L₂-methodsand certain weight functions. He was able to demonstrate existence (in the sense treated in § 8.4 below) of forms of type (p, q) in cases where the Levi form ((1.2) below) either has at least q + 1 negative eigenvalues or at least n - q positive eigenvalues. This is a much less restrictive condition on b M than our condition of pseudo-convexity.