Analytic structure of $q$-pseudoconcave subsets of continuous graphs

Abstract

The aim of this article is to find under which conditions there exists a foliation by $n$-dimensional complex manifolds on a closed subset $Z\subset\mathbb{C}^N$, which is locally a continuous graph over a closed subset of $\mathbb{C}^n\times\mathbb{R}$. We prove that such foliation does exist if $Z$ is $q$-pseudoconcave, for any $q\geq n$. We also prove that this bound on the index of pseudoconcavity is sharp. Namely, we construct many $q$-pseudoconcave subsets of $\mathbb{C}^N$, for $q<n$, which are smooth graphs over closed subsets of $\mathbb{C}^n\times\mathbb{R}$ but with no analytic structure. As an application, we show that an $n$-pseudoconcave set sitting in a subset of $\mathbb{C}^N$, which is locally a differentiable graph over a domain of $\mathbb{R}^{2n+1}$, is foliated by $n$-dimensional complex manifolds.