Three Problems in Complex Analysis

项目负责人：David catlin，主要研究代数几何、微分几何和偏微分方程中的三个独立问题。第一个问题与E. Artin经典定理的复变量类比有关，该定理指出，n个实变量的非负多项式可以写成有理函数的平方和。该提案旨在确定n个复变量中给定的非负多项式何时可以写成两个函数的商，其中每个函数都是全纯多项式的平方和。一个猜想被陈述，作为一种特殊的情况，可能会导致一个建设性的版本的阿廷定理。第二个问题与Kuranishi问题有关，该问题寻求涉及Levi形式的几何条件，这意味着CR流形的给定局部邻域可以嵌入到复欧几里德空间中。该提案描述了一种可能的方法来解决当相关的CR结构是最小平滑时找到这样一个嵌入的问题。第三个问题是在无穷型域的边界下，求复诺伊曼问题的紧性或亚椭圆性的充分条件。众所周知，并非所有数学结果都能立即应用于所谓的现实世界问题。有时，一个重要命题的证明依赖于抽象的方法，而这些方法不容易应用于需要明确构造的更具体的情况。对于第一个问题，提议者正在检查代数中的一个著名定理，该定理使用抽象方法来解释为什么某些多项式总是非负的。为了使这个定理在现实世界中得到应用，找到这个定理的构造性证明是非常重要的。这个提议的第二部分与几何学中的一个问题有关，在这个问题中，一个表面实际上不是光滑的，它有尖角或扭曲。这样的问题对于应用来说也很重要，因为在与物体形状有关的问题中，物体很少是完全光滑的。虽然所研究的问题在一定程度上是理论性的，但在一个问题上开发的技术往往也可以用于相关领域。

AbstractAward: DMS-9971935Principal Investigator: David CatlinThe proposal is concerned with three separate problems inalgebraic geometry, differential geometry and in partialdifferential equations.The first problem is concerned with acomplex variables analog of the classical theorem of E. Artin,which states that a non-negative polynomial in n real variablescan be written as a sum of squares of rational functions. Theproposal seeds to determine exactly when a given nonnegativepolynomial in n complex variables can be written as a quotient oftwo functions, each of which is the sum of squares of holomorphicpolynomials. A conjecture is stated, which as a special casewould likely lead to a constructive version of Artin's theorem.The second problem is concerned with the Kuranishi problem, whichseeks geometric conditions involving the Levi form that implythat a given local neighborhood of a CR manifold can be embeddedinto complex Euclidean space. The proposal describes a possibleapproach to the problem of finding such an embedding when theassociated CR structure is minimally smooth. The third problemis concerned with finding sufficient conditions for eithercompactness or hypoellipticity of the complex Neumann problem inthe case when the boundary of the domain is of infinite type.It is well-known fact that not all results in mathematics can beimmediately applied to so-called real world problems. Sometimesthe proof of an important statement depends on abstract methodswhich cannot be easily applied in more concrete situations whichcall for an explicit construction. For the first problem, theproposer is examining a well-known theorem in algebra which usesabstract methods to explain why certain polynomials are alwaysnonnegative. In order for this theorem to be used in the realworld, it would be very important to find a constructive proof ofthis theorem. The second part of the proposal has to do with aproblem in geometry in which a a surface in effect is not smoothand has sharp corners or twists built into it. Such problems arealso important for applications because in problems having to dowith the shapes of objects, it is rarely the case that theobjects are completely smooth. Although the problem that isstudied is somewhat theoretical, it often happens that techniquesdeveloped in one problem can often be used in related areas aswell.