CR-流形的复配边性与可嵌入性
Complex cobordism and embeddability of CR-manifolds
摘要：
本文研究了紧的、三维的、严格伪凸的Cauchy-Riemann流形之间的复配边问题。设复配边由复2-流形X给出，X具有一个伪凸和一个伪凹端。我们回答以下问题。X的伪凸端的可嵌入性和伪凹端的可嵌入性之间的关系是什么？X的伪凸端上的CR-函数是否都扩张为X内部的全纯函数？证明了严格伪凸Cauchy-Riemann三维流形的可嵌入性不是复协边不变量。我们证明了一个新的现象：在X上的伪凸端存在CR-函数，它们不延伸到全纯函数。我们还证明了CR-函数从伪凸端的可拓性是必要的，但不足以在复配边下保持可嵌入性。
英文摘要：
This paper studies complex cobordisms between compact, three dimensional, strictly pseudoconvex Cauchy-Riemann manifolds. Suppose the complex cobordism is given by a complex 2-manifold X with one pseudoconvex and one pseudoconcave end. We answer the following questions. What is the relation between the embeddability of the pseudoconvex end and the embeddability of the pseudoconcave end of X? Do all CR-functions on the pseudoconvex end of X extend to holomorphic functions on the interior of X? We prove that embeddability of a strictly pseudoconvex Cauchy-Riemann 3-manifold is not a complex-cobordism invariant. We show that a new phenomenon occurs: there are CR-functions on the pseudoconvex end that do not extend to holomorphic functions on X. We also show that the extendability of the CR-functions from the pseudoconvex end is necessary but not sufficient for embeddability to be preserved under complex cobordisms.
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