论文标题
$ c^2 $边界的全球纽兰德 - 尼伦贝格定理
Global Newlander-Nirenberg theorem for domains with $C^2$ boundary
论文作者
论文摘要
Newlander-Nirenberg Theorem说，正式的整合复杂结构在局部等同于复杂欧几里得空间中的标准复杂结构。在本文中，我们考虑了在$ c^2 $严格的伪convex边界的存在下，纽兰德 - 尼伦堡定理的两个自然概括。当给定的正式集成的复杂结构$ x $是在关闭有限的严格伪有元素域中定义的，该域具有$ C^2 $边界$ d \ subset \ subset \ mathbb {c}^n $时，我们显示了在$ \ + edline complends $ x Complement $ x Complement $ x Complement $ x Complement $ x Complement $ x Complement $ x Complement $ x Complement $ x Complement $ x Complement $ x Complecter to $ x $ complecter to $ x $ x $ x $中的存在的存在。此外，我们表明，在$ \ partial d $的小$ c^2 $扰动下，这种亲密关系是稳定的。结果，当给定的正式集成的复杂结构被定义在$ C^2 $真正的hypersurface $ m \ subset \ subset \ mathbb {c}^n $中的单方面邻域中时，我们证明存在局部单方面的holomorphic坐标系统的存在，规定$ M $是严格涉及给定复杂结构的$ M $。当结构平滑时，我们还会获得结果。
The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander-Nirenberg theorem under the presence of a $C^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure $X$ is defined on the closure of a bounded strictly pseudoconvex domain with $C^2$ boundary $D\subset \mathbb{C}^n$, we show the existence of global holomorphic coordinate systems defined on $\overline{D}$ that transform $X$ into the standard complex structure provided that $X$ is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small $C^2$ perturbation of $\partial D$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a $C^2$ real hypersurface $M\subset \mathbb{C}^n$, we prove the existence of local one-sided holomorphic coordinate systems provided that $M$ is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.
