A Yang--Mills theoretic proof of the Newlander--Nirenberg theorem

Comments on the Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem says that a necessary and sufficient condition for the complex coordinates associated with a given almost complex structure tensor $I_M{}^N$ to exist is the vanishing of the Nijenhuis tensor ${\cal N}_{MN}{}^K$. In the first part of the paper, we give a simple explicit proof of this fact. In the second part, we discuss a supersymmetric interpretation of this theorem. ${\it (i)}$ The condition ${\cal N}_{MN}{}^K = 0$ is necessary for a certain $N=1$ supersymmetric mechanical sigma models to enjoy $N=2$ supersymmetry. ${\it (ii)}$ The sufficiency of this condition…

具有广义边界的B复流形。I. 新兰德-尼伦伯格定理

B-complex manifolds with generalized corners. I. Newlander-Nirenberg Theorems

本文将复流形推广到具有边界的流形，通过b切丛上的复结构证明了新兰德-尼伦伯格型定理，展示了具有广义边界的B复流形在边界层中的结构，并给出了成为Kato-Nakayama空间的充要条件。

Global Newlander-Nirenberg theorem for domains with $C^2$ boundary

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…

复 $b$-流形的Newlander-Nirenberg定理

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。

Sub-Hermitian Geometry and the Quantitative Newlander-Nirenberg Theorem

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…

关于全局新兰德-尼伦伯格定理的1/2估计

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。