通过拓扑同位素构造 Stein 曲面
Creating Stein surfaces by topological isotopy
AI总结：
本文结合 Freedman 拓扑与 Eliashberg 全纯理论，证明复曲面中驯服拓扑嵌入的 2-复形经小拓扑同位素后可作为不可数多个 Stein 正则邻域的交集，并借此研究拓扑伪凸性与伪凹性。
摘要：
我们将 Freedman 的拓扑学与 Eliashberg 的全纯理论相结合，在复曲面中构造 Stein 邻域系统，并利用这些系统研究各种凸性与凹性概念。在复曲面中，每个驯服、拓扑嵌入的 2-复形 K，经过 C^0-小拓扑环境同位素后，都是一个不可数嵌套 Stein 正则邻域族的交集，这些邻域都相对于 K 拓扑环境同位素，但常常实现不可数个微分同胚类型。这些类型来自一个拓扑映射柱的 Cantor 集层次。这些邻域的边界是仅拓扑嵌入的 3-流形，但仍满足一种伪凸性概念。这类 3-流形具有通常光滑意义下严格伪凸超曲面的一些基本性质，但它们远为普遍。与之互补的拓扑伪凹性概念由同胚于 R^4 的不可数个微分同胚类型实现。
英文摘要：
We combine Freedman's topology with Eliashberg's holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded 2-complex K in a complex surface, after C^0-small topological ambient isotopy, is the intersection of an uncountable nested family of Stein regular neighborhoods that are all topologically ambiently isotopic rel K, but frequently realize uncountably many diffeomorphism types. These arise from the Cantor set levels of a topological mapping cylinder. The boundaries of the neighborhoods are 3-manifolds that are only topologically embedded, but still satisfy a notion of pseudoconvexity. Such 3-manifolds share some basic properties of hypersurfaces that are strictly pseudoconvex in the usual smooth sense, but they are far more common. The complementary notion of topological pseudoconcavity is realized by uncountably many diffeomorphism types homeomorphic to R^4.
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