Mathematical Sciences: Applications of the d-bar Neumann Problem to the Study of Weakly Pseudoconvex Domains

这项工作的中心主题是几何问题， 几个复杂的变量 三个问题将在 目前的项目，第一个是Kuranishi问题。 这里 一个是对结构相似的真实的流形感兴趣 到复杂空间中域的边界。 争论的焦点是是否 或非光滑伪凸C-R流形允许嵌入。 的 这个问题的分析方面是解决一个复杂的 混合边界条件偏微分方程 如果 研究按预期进行，嵌入结果应 可以用Levi的前两个特征向量表示 形式的流形（他们必须是积极的）。 另一项类似性质的工作涉及 伪凸基本一阶d-bar-Neumann问题 域. 一个是有兴趣找到解决方案， 正交域上的非齐次d杆方程 该域上的全纯函数空间。 问题 解的全局正则性不仅取决于 域的几何形状，而且还取决于度量（其确定 体积元素）。 将开展工作， 光滑有界伪凸空间的全局正则性条件 域中的曲率不等式，必须是 满足于公制。 本研究的第三个要素涉及自同构或 有界域的自映射 贝尔和平丘克最近的工作 基本上把所有的伪凸域分为两类- 具有非紧自同构群的维复空间（ 域也必须是有限类型）。 一些相关 问题仍然开放，将在此期间进行分析。 事业。

The central theme of this work is geometric problems in several complex variables. Three questions will be addressed in the current project, the first being the Kuranishi problem. Here one is interested in real manifolds which have structure similar to boundaries of domains in complex space. At issue is whether or not smooth pseudoconvex C-R manifolds admit embeddings. The analytic side of this question is one of solving a complex partial differential equation with mixed boundary conditions. If the research proceeds as predicted, the embedding result should be expressible in terms of the first two eigenvectors of the Levi form for the manifold (they must be positive). Another line of work of a similar nature concerns the fundamental first order d-bar-Neumann problem on pseudoconvex domains. One is interested in finding solutions of the inhomogeneous d-bar equation on a domain which are orthogonal to the space of holomorphic functions on that domain. The question of global regularity of the solutions depends not only on the geometry of the domain but also on the metric (which determines the volume element.). Work will be done in establishing conditions for global regularity on smoothly bounded pseudoconvex domains in terms of curvature inequalities which must be satisfied by the metric. The third element of this research concerns automorphisms or self-maps of bounded domains. Recent work of Bell and Pinchuk has essentially classified all pseudoconvex domains in two- dimensional complex space with noncompact automorphism group (the domains must also be of finite type). A number of related questions remain open which will be analyzed during this undertaking.