Liding Yao
We give an example of $C^k$-integrable almost complex structure that does not admit a corresponding $C^{k+1}$-complex coordinate system.
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The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander-Nirenberg theorem with parameter. The first extends the Newlander-Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster's proof for the Newlander-Nirenberg theorem. The second concerns a version of Nirenberg's complex Frobenius theorem and its proof yields a result with a mild loss of regularity.
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 $X$ in…
Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields so that there is a complex structure on $M$ with respect to which the vector fields are $T^{0,1}$. In this paper, we give intrinsic, diffeomorphic invariant, necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure (i.e., smooth, real analytic, or have Zyg…
Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces
本文基于自由解析sheaves在最大和齐性空间上，证明了相对Riemann-Hilbert定理在考虑torsion时仍成立，并推导出Newlander-Nirenberg型定理，揭示了tame flat连接与局部系统及基本群表示的对应关系。
The Newlander-Nirenberg theorem for complex $b$-manifolds
本文证明复 $b$-流形具有仅依赖于维数的局部模型，这相当于复 $b$-流形的Newlander-Nirenberg定理，证明利用了Mendoza的“形式局部不变量”结果和奇异坐标变换，结合经典Newlander-Nirenberg定理及Catlin对伪凸边界复流形的推广。
On $1/2$ estimate for global Newlander-Nirenberg theorem
本文研究了在严格伪凸域上，近复结构在一定正则性条件下是否存在全局微分同胚转化为标准复结构的问题，并给出了1/2估计的定量结果。
Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds
本文在边界 Levi 形式满足一定特征值条件的复流形区域上，利用同伦公式和 Nash-Moser 迭代证明：充分接近原复结构的形式可积近复结构可由一个微分同胚拉回为复结构，从而得到全局 Newlander-Nirenberg 定理。
We give a characterization of the complex Euclidean space C^n, proving a global Newlander-Nirenberg's theorem on the representability of complex structures close to the standard complex structure on C^n.
The Newlander-Nirenberg Theorem for principal bundles
本文研究主丛上Hölder类丛殆复结构的可积性，构造可积性障碍形式，证明其为零等价于局部存在高阶伪全纯截面并给出全纯约化，结果将用于全纯丛模空间研究。
We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.
We show that on the unit ball of a certain separable Banach space there is a smooth delbar-closed (0,1)-form which is not locally delbar-exact. Further, the Dolbeault isomorphism theorem does not generalize to arbitrary Banach spaces. Lastly, the Newlander-Nirenberg theorem does not generalize to arbitrary smooth integrable almost complex Banach manifolds.
As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in such a way that the structure is locally the span of $\frac{\partial}{\partial t_1},\ldots, \frac{\partial}{\partial t_r},\frac{\partial}{\partial \overline{z}_1},\ldots, \frac{\partial}{\partial \overline{z}_n}$; where $\mathbb{R}^r\times \mathbb{C}^n$ has coordinates $(t_1,\ldots, t_r, z_1,\ldots, z_n)$. In this paper, we give optimal regularit…
We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on previous recent work of the authors.
Local and global blow-downs of transport twistor space
本文研究传输扭量空间的退化复结构，通过构造全纯 blow-down 映射，为常曲率度量提供整体解、任意度量提供局部解，并证明了退化情形的 Newlander--Nirenberg 定理。
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