Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds
Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds
复制标题
三维严格伪凸 CR 流形的变形和嵌入
DOI：
10.1007/s00208-023-02658-y
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发表时间：
2020
影响因子：
1.4
通讯作者：
P. Ebenfelt
中科院分区：
文献类型：
--
作者：
Sean N. Curry;P. Ebenfelt
$\mathbb{C}^2$中紧致严格伪凸超曲面$M$的CR结构的抽象变形用$M$上的复变函数编码。与高维情形形成鲜明对比的是，3 $维CR结构的自然可积性条件是空洞的，紧致严格伪凸超曲面$M\subseteq \mathbb{C}^2 $的一般变形即使在$\mathbb{C}^N $中对任何$N$也是不可嵌入的。一个基本的（也是困难的）问题是刻画$M \subseteq \mathbb{C}^2$上的复函数何时会在$\mathbb{C}^2$内产生$M$的实际变形。本文研究了嵌入CR 3 $-流形上变形族的可嵌入性以及S ^3 $上可嵌入CR结构空间的结构。证明了标准CR 3 $-球面的可嵌入变形空间是C^{\infty}（S^3，\mathbb {C}）$在原点附近的Frechet子流形.我们建立了一个修改后的版本的Cheng-Lee切片定理，在其中我们能够精确地描述切片中的可嵌入变形（在球谐函数方面）。我们还引入了一个规范的家庭嵌入变形和相应的嵌入从任何无穷小嵌入变形的单位球在$\mathbb{C}^2 $。
Abstract deformations of the CR structure of a compact strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ are encoded by complex functions on $M$. In sharp contrast with the higher dimensional case, the natural integrability condition for $3$-dimensional CR structures is vacuous, and generic deformations of a compact strictly pseudoconvex hypersurface $M\subseteq \mathbb{C}^2$ are not embeddable even in $\mathbb{C}^N$ for any $N$. A fundamental (and difficult) problem is to characterize when a complex function on $M \subseteq \mathbb{C}^2$ gives rise to an actual deformation of $M$ inside $\mathbb{C}^2$. In this paper we study the embeddability of families of deformations of a given embedded CR $3$-manifold, and the structure of the space of embeddable CR structures on $S^3$. We show that the space of embeddable deformations of the standard CR $3$-sphere is a Frechet submanifold of $C^{\infty}(S^3,\mathbb{C})$ near the origin. We establish a modified version of the Cheng-Lee slice theorem in which we are able to characterize precisely the embeddable deformations in the slice (in terms of spherical harmonics). We also introduce a canonical family of embeddable deformations and corresponding embeddings starting with any infinitesimally embeddable deformation of the unit sphere in $\mathbb{C}^2$.