On an $n$-manifold in ${\mathbf{C}}^{n}$ near an elliptic complex tangent

In this paper, we study the local biholomorphic property of a real $n$-manifold $M\subset\mathbf C^n$ near an elliptic complex tangent point $p\in M$. In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy $\widetilde{M}$ of $M$ near $p$, first considered in a paper of Bishop. When $M$ is a $C^{\infty}$-smooth submanifold, using a result established by Kenig-Webster, we show that near $p$, $\widetilde{M}$ is a smooth Levi-flat $(n+1)$-manifold with a neighborhood of $p$ in $M$ as part of its $C^{\infty}$ boundary. Moreover, near $p$, $\widetilde{M}$ is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to $M$. This result was proved in the previous work of Kenig-Webster when $n=2$. When $M$ is real analytic, we show that $\widetilde{M}$ is real analytic with a neighborhood of $p$ in $M$ as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When $n=2$ or when $n>2$ but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author's joint work with Krantz.