论文标题
三维严格的假子宫vvex cr歧管的变形和嵌入
Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds
论文作者
论文摘要
紧凑型pseudoconvex hypersurface $ m $ in $ \ mathbb {c}^2 $的cr结构的抽象变形由$ m $上的复杂函数编码。与较高维度的情况形成鲜明对比的是，$ 3 $维的CR结构的自然可合转条件是空虚的，并且紧凑的严格pseudoconvex hypersurface $ m \ subseteq \ subseteq \ mathbb {c}^2 $甚至在$ \ mathbb {c} c}^n $中也无法嵌入。一个基本（且困难的）问题是表征$ M \ subseteq \ Mathbb {C}^2 $的复杂函数时，会导致$ \ Mathbb {C}^2 $的实际变形。在本文中，我们研究了给定嵌入式CR $ 3 $ manifold的变形家庭的嵌入性，以及$ s^3 $上可嵌入的CR结构空间的结构。我们表明，标准Cr $ 3 $ -sphere的可嵌入变形空间是$ C^{\ infty}（s^3，\ Mathbb {C}）$的Frechet submanifold。我们建立了Cheng-Lee Slice定理的修改版本，在该定理中，我们能够精确表征切片中的可嵌入变形（以球形谐波）。我们还引入了一个可嵌入变形的典型家族和相应的嵌入，从$ \ mathbb {c}^2 $中的任何无限嵌入式嵌入式变形开始。
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 $\mathbb{C}^2$.
