Complex Manifolds In $Q$-Convex Boundaries

TL;DR

This paper extends the understanding of q-convex boundaries by proving tangent properties and subelliptic estimates, generalizing previous pseudoconvex results and analyzing the implications for the di-bar-Neumann problem.

Contribution

It generalizes Diederich-Fornaess's results from pseudoconvex to q-convex domains and links the absence of complex q-manifolds to subelliptic estimates for q-forms.

Findings

bΩ is tangent of infinite order to complexification of its submanifolds

Levi-form rows are (1/2)-subelliptic multipliers for the di-bar-Neumann problem

Absence of complex q-manifolds implies subelliptic estimates

Abstract

We consider a smooth boundary b\Omega which is q-convex in the sense that its Levi-form has positive trace on every complex q-plane. We prove that b\Omega is tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type. This generalizes Diederich-Fornaess [Annals 1978] from pseudoconvex to q-convex domains. We also readily prove that the rows of the Levi-form are (1/2)-subelliptic multipliers for the di-bar-Neumann problem on q-forms (cf. Ho [Math. Ann. 1991]). This allows to run the Kohn algorithm of [Acta Math. 1979] in the chain of ideals of subelliptic multipliers for q-forms. If b\Omega is real analytic and the algorithm stucks on q-forms, then it produces a variety of holomorphic dimension q, and in fact, by our result above, a complex q-manifold which is not only tangent but indeed contained in b\Omega. Altogether, the absence of complex q-manifolds in b\Omega produces a subelliptic estimate on q-forms.