• Journal of the Korean Mathematical Society
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• v.54 no.6
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• pp.1787-1799
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• 2017

In this paper, we prove basic a priori estimate for the ${\bar{\partial}}$-Neumann problem on an annulus between two pseudoconvex submanifolds of a Stein manifold. As a corollary of the result, we obtain the global regularity for the ${\bar{\partial}}$-problem on the annulus. This is a manifold version of the previous results on pseudoconvex domains.

• East Asian mathematical journal
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• v.15 no.1
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• pp.147-151
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• 1999

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.563-575
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• 2003

We show that if a connected Stein manifold M of dimension n has the holomorphic automorphism group Aut(M) isomorphic to $Aut(C^k {\times}(C^*)^{n - k})$ as topological groups, then M itself is biholomorphically equivalent to C^k{\times}(C^*)^{n - k}$. Besides, a new approach to the study of U(n)-actions on complex manifolds of dimension n is given.

• Journal of the Korean Mathematical Society
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• v.47 no.4
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• pp.675-689
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• 2010

For a closed symplectic 4-manifold X, let $Diff_0$(X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of $Diff_0$(X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {$n_1$, $n_2$, $\ldots$, $n_k$} and any non-negative integer m, there exists a closed symplectic (or K$\ddot{a}$hler) 4-manifold X with $b_2^+$ (X) > m such that the homologies $H_i$ of the quotient space $Diff_0$(X)/Symp(X) over the rational coefficients are non-trivial for all odd degrees i = $2n_1$ - 1, $\ldots$, $2n_k$ - 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Mati$\acute{c}$.