流形上的调和形式

流形上的调和形式是研究流形的几何与拓扑性质的重要工具。我们拟从如下三个方面开展研究：四维殆复流形上的J-反变上同调群、完备流形上L^2调和形式、奇数维辛流形上的上同调群。拟研究如下问题：(1) 四维闭辛流形上在最一般情形下的Donaldson问题和Donaldson型Calabi-Yau方程解的存在性；(2) 完备辛流形上L^2调和形式的精细Kato不等式以及球面中完备非紧子流形的消灭定理和L^2调和1-形式所构成空间的维数的有限性；(3) 五维切触流形上基本Laplace算子的特性，特别是关于J-反变基本2-形式构成的集合的刻画和J-反变基本上同调群维数的稳定性等；切触流形的基本上同调群的分解；建立紧五维K-切触流形上的广义Calabi-Yau方程并探讨其解的存在性。

Harmonic forms on manifolds is an important tool to study geometric and topological properties of manifolds. We plan to study from the following three aspects: J-invariant cohomological groups on four-dimenisonal almost complex manifolds，L^2 harmonic forms on complete manifolds and cohomologial groups on odd dimenisonal symplectic manifolds. We will discuss the following problems: (1) Donaldson question with most generalized case and the existence of solutions of Donaldson-type Calabi-Yau equation on closed four-dimensional symplectic manifolds;(2) refined Kato inequalities for L^2 harmonic forms on complete symplectic manifolds, vanishing theorems and finiteness of dimension of the space of L^2 harmonic 1-forms on complete noncompact submanifolds in spheres; (3) properties of basic Laplacian operations on five-dimenisonal contact manifolds, in particular, description of the space of J-invariant basic 2-forms and stability of dimension of J-invariant basic cohomology groups; decomposition of basic cohomology groups on contact manifolds; set up generalized Calabi-Yau equation on five-dimensional K-contact manifolds and discuss the existence of its solutions.