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

Author: Xiaojun Huang
Journal: J. Amer. Math. Soc. 11 (1998), 669-692
MSC (1991): Primary 32F25, 32D05
MathSciNet review: 1603854
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Abstract: 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.

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Xiaojun Huang
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Received by editor(s): August 8, 1997
Received by editor(s) in revised form: February 9, 1998
Additional Notes: The author was supported in part by NSF DMS-9500881 and an NSF postdoctoral fellowship
Article copyright: © Copyright 1998 American Mathematical Society