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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On an $n$-manifold in $\mathbf {C}^n$ near an elliptic complex tangent
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by Xiaojun Huang PDF
J. Amer. Math. Soc. 11 (1998), 669-692 Request permission

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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Additional Information
  • Xiaojun Huang
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
  • Email: huangx@math.rutgers.edu
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 669-692
  • MSC (1991): Primary 32F25, 32D05
  • DOI: https://doi.org/10.1090/S0894-0347-98-00265-3
  • MathSciNet review: 1603854