On three-dimensional Cauchy-Riemann manifolds
On three-dimensional Cauchy-Riemann manifolds
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DOI：
10.1090/s0894-0347-1992-1157290-3
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发表时间：
1992-01
影响因子：
3.9
通讯作者：
L. Lempert
中科院分区：
文献类型：
--
作者：
L. Lempert
C2的几乎复张量(“乘以i”)。这些子空间的集合Hp M形成一个丛Hm c TM，称为水平丛或接触丛。(实际上，M的严格伪凸性意味着平面场{HPM}是非退化的，即定义了一种接触结构。)然后C2的几乎复张量限制为丛自同态J：Hm-Hm使得J2=-id。丛HMcTM和这个自同态J一起定义了M的CR结构。稍微更一般地，设D=DUM是一个四维紧致流形，边界是M，内部是D，D上有一个光滑的几乎复张量，这个张量在D上可积。如果D是严格伪凸的--即在任意p-EM的邻域内有一个光滑的严格多次调和函数u，它在D上是负的，在M上是0的，但Du$0在M上--那么M继承了D的一个严格伪凸CR结构。这将我们引向抽象的定义。严格伪凸(三维)CR流形是一个紧致流形M(无边界)，维M=3，具有接触结构HM={HPM：P E M}cTM和HM的一个自同态J使得J2=-id.(在整篇文章中，除非另有说明，否则我们将处理无限可微对象。因此，假设M、HM、J在这个意义上是光滑的。)此外，一个可微函数f：m-？C是CR函数，如果对任何p E M，限制dF IHM：HPM-Tf(p，)C
of the almost complex tensor of C2 ("multiplication by i "). The collection of these subspaces Hp M forms a bundle HM c TM called the horizontal or contact bundle. (Indeed, strict pseudoconvexity of M implies that the plane field {HpM} is nondegenerate, i.e. defines a contact structure.) The almost complex tensor of C2 then restricts to a bundle endomorphism J: HM -HM such that J2 = -id. The bundle HM c TM together with this endomorphism J defines the CR structure of M. Slightly more generally, let D = D U M be a four dimensional compact manifold with boundary M, interior D, endowed with a smooth almost complex tensor which is integrable on D. If D is strictly pseudoconvex-i.e., in a neighborhood of any p E M there is a smooth strictly plurisubharmonic function u, negative on D, 0 on M, but du $ 0 on M-then M inherits a strictly pseudoconvex CR structure from D. In this case we shall say that M bounds a strictly pseudoconvex surface. This leads us to the abstract definition. A strictly pseudoconvex (threedimensional) CR manifold is a compact manifold M (without boundary), dim M = 3, endowed with a contact structure HM = {HpM: p E M} c TM and an endomorphism J of HM such that J2 = -id. (Throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects. Thus M, HM, J are assumed to be smooth in this sense.) Furthermore, a differentiable function f: M -? C is a CR function if for any p E M the restriction df IHM: HpM -Tf(p,)C