The null space of the $\bar\partial $-Neumann operator

Abstract

Let Q be a complex analytic manifold of dimension n with a hermitian metric and C-infinity boundary, and let rectangle = deltadelta* + delta* delta be the self-adjoint delta-Neumann operator on the space L-0,q(2) (Omega) of forms of type (0, q). If the Levi form of deltaOmega has everywhere at least n - q positive or at least q+ I negative eigenvalues, it is well known that Ker rectangle has finite dimension and that the range of rectangle is the orthogonal complement. In this paper it is proved that dim Ker rectangle = infinity if the range of rectangle is closed and the Levi form of deltaOmega has signature n - q - 1, q at some boundary point. The starting point for the proof is an explicit determination of Ker rectangle when Omega subset of C-n is a spherical shell and q = n - 1. Then Ker rectangle has n independent multipliers; this is only true for shells Omega subset of C-n bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on Ker rectangle when the range of 0 is closed, at points on deltaOmega where the Levi form is negative definite, q = n - 1. Crude bounds are also given when the signature is n - q - 1, q with 1 less than or equal to q less than or equal to n - 1. (Less)