复Finsler几何中的曲率和拓扑

Finsler geometry is a more general differential geometry than the Riemannian geometry, under the initiation of S.S.Chern, the global Finsler geometry has gained a substantial development in recent decades. This project intends to study the curvature properties of complex Finsler geometry and use it to characterize the geometry and topology of complex Finsler manifolds, and we will study some problems related to the several complex variables and complex geometry. Firstly, we will study the connections and curvatures in complex Finsler geometry, and then we will give some applications of them. Using a strongly pseudoconvex complex Finsler metric, we will give the reasonable definitions of some important curvatures such as flag curvature and bisectional curvature in complex Finsler geometry. At the same time, the geometrical and topological properties of complex Finsler manifold are studied, including the study of the global properties of manifold by the Morse index theory and comparison theorems. Secondly, we will generalize the Bochner technique on the induced vector bundles and get some vanishing theorems in the complex Finsler geometry. Finally, this project will study some applications of complex Finsler metrics in several complex variables and complex geometry, including the criterions of the Stein manifolds by the curvatures of complex Finsler geometry and the Schwarz lemmas in complex Finsler geometry.

Finsler几何是比Riemann几何更一般的微分几何，在陈省身先生的倡导下，近几十年来取得了实质性进展。本项目拟系统研究复Finsler几何中的曲率性质并用它来刻画复Finsler流形的几何和拓扑等整体性质，同时结合多复变与复几何相关问题进行研究。首先我们将研究复Finsler几何中的联络和曲率，并给出它们的一些应用。我们将用强拟凸复Finsler度量给出复Finsler几何中的旗曲率和双截曲率等重要曲率的合理定义。同时研究复Finsler流形上的几何与拓扑性质，包括用Morse指标理论和比较定理来研究流形的整体性质。其次我们将在诱导向量丛上推广Bochner技巧并得到复Finsler几何中的消灭定理。最后，本项目将研究复Finsler度量在多复变和复几何中的一些应用，包括用复Finsler几何中的曲率给出Stein流形的一些刻画和研究复Finsler几何中的Schwarz引理。

本项目主要通过研究复Finsler度量的曲率，给出复Finsler流形上的几何和拓扑性质。具体来说，我们得到如下几个方面结果：（1）我们给出了Kahler-Finsler流形上全纯双截曲率的几何解释，并证明了Kahler-Finsler流形上的Frankel相交定理和Kahler-Finsler曲面上的Frankel猜测；（2）我们给出了2个复Finsler流形之间的全纯映射的刚性定理，并讨论了复Finsler几何中的Schwarz引理。同时用复Finsler度量给出双曲流形的一个新刻画；（3）我们得到复Finsler几何中的几类消没定理，并给出了复Finsler几何上的一些应用。