Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

Abstract

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C n is an obstruction to compactness of the ∂-Neumann operator on , provided that at some point of M, the Levi form of b has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. (2000): 32W05 Mathematics Subject Classification

On noncompactness of the ∂‾-Neumann problem on pseudoconvex domains in C3

“…(d) Fu and Straube showed in [FS01] that on bounded locally convexifiable domains (in any dimension) N q is compact if and only if there are no qdimensional complex manifolds in the boundary (see also section 4.9 of [Str10]). (e) S ¸ahutoglu and Straube proved failure of compactness of N q for smooth pseudoconvex domains in C n with a q-dimensional manifold M in the boundary, under the additional assumption that at a point of M the boundary is strictly pseudoconvex in the directions transverse to M (see [cSgS06] for the case q = 1, and [Sah06] or Theorem 4.21 of [Str10] for the general case). Notice that if q = n − 1 this hypothesis is empty and the problem is completely solved.…”

“…The rest of the paper is devoted to the proof of Theorem 1. It follows the "plan" of [cSgS06] and [FS98], aiming to insert a smaller domain inside Ω that is tangent to bΩ along D and has a product structure in appropriate coordinates. Morally, it is this product structure that causes the failure of compactness, but one must choose the smaller domain in such a way that its boundary has a sufficiently high order of contact with bΩ, in order to transfer the non-compactness to the original domain Ω.…”

“…Morally, it is this product structure that causes the failure of compactness, but one must choose the smaller domain in such a way that its boundary has a sufficiently high order of contact with bΩ, in order to transfer the non-compactness to the original domain Ω. Under the assumptions of [cSgS06], the minimal order of contact 2 is enough, and the construction boils down to a local structure lemma for domains with complex manifolds in the boundary which, in the q = 1 case, follows from a result of Bedford and Fornaess (Lemma 1 of [BFs81]).…”

Geometric Sufficient Conditions for Compactness of the Complex Green Operator

“…Because bΩ contains no analytic discs, S X,P contains points ζ arbitrarily close to P such that L(X(ζ), X(ζ)) > 0 ( [25], Lemma 3). Because 0 < L(X(ζ), X(ζ)) ≤ Cλ 0 (ζ), ζ is a strictly pseudoconvex point.…”

The Restriction Operator on Bergman Spaces

“…These investigations were continued by Mijoung Kim in her thesis [Kim03] and in [Kim04], where U is said to be "fat" or "thick" in Ω if R U is not compact. Further extensions of these ideas may be found in [ŞS06,Dal18].…”