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

Abstract

AbstractIn 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.KeywordsNEUMANN ProblemOrthogonal BasisGeodesic DistanceLevi FormCoordinate PatchThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.