A criterion for the neumann type problem over a differential complex on a strongly pseudo convex domain
A criterion for the neumann type problem over a differential complex on a strongly pseudo convex domain
复制标题
强伪凸域上微分复形上的诺伊曼型问题的判据
DOI：
10.1007/bf01456959
复制
发表时间：
1983
影响因子：
1.4
通讯作者：
T. Akahori
中科院分区：
文献类型：
--
作者：
T. Akahori
本文的目的是构造四维情形的强伪凸域族。设N是复流形，f2是其具有强伪凸边界bf2的相对紧子域.在（2）中，在dimcN> 5的情况下，建立了强伪凸域族（这通过构造在边界处具有适当行为的新Neumann算子来解决）。另一方面，对于边界情况，我们已经知道在dimRbt2 = 2n-1> 7，也就是说n> 4的情况下，CR结构族的存在性[cf.①]。因此，很自然地，我们可以用同样的方法构造n = 4的强伪凸域的Neumann族，即构造一个新的具有适当性质的Neumann型算子[在n> 5的情况下，这在（2）中通过直接计算得到了证明]。然而，在本文中，我们使情况更一般，并给出了一个标准（定理4。1）。作为这个准则的应用，我们将得到满足适当边界条件的新Neumann型算子（推论6.2），从而在n> 4的情况下构造了强伪凸域的Neumann族.这种方法的优点是，我们的证明比前一个简单得多[cf. (2)]同时，对于n> 4的情形，我们可以给出新Neumann算子的存在性.这项工作开始于哥伦比亚大学，并在波恩的马克斯-普朗克数学研究所完成。
The motivation of this paper is to construct the versal family of strongly pseudo convex domains for the four dimensional case. Let N be a complex manifold and f2 be its relative compact subdomain with strongly pseudo convex boundary bf2. In (2), the versal family of strongly pseudo convex domains was established in the case dimcN> 5 (this was solved by constructing the new Neumann operator with a suitable behavior at the boundary). On the other hand, for the boundary case we have already known the existence of the versal family of CR structures for the case dimRbt2= 2n-1> 7, that is to say, n> 4 [cf.(1)]. Therefore it is natural to try to construct the versal family of strongly pseudo convex domains for n= 4 by the same method, ie, to construct a new Neumann type operator with a suitable property [in the case n> 5, this was shown by a direct computation in (2)]. However, in this paper we make the situation more general and give a criterion (Theorem4. 1). As an application of this criterion, we will obtain the new Neumann type operator satisfying a suitable condition on the boundary (Corollary 6.2), which leads to the construction of the versal family of strongly pseudo convex domains for the case n> 4. The advantage of this approach is that our proof is much simpler than the former one [cf.(2)] and at the same time we can give the existence of the new Neumann operator for the case n> 4. This work was initiated in the Columbia Univ. and completed in Max-Planck-Institut f'tir Mathematik in Bonn.