酉不变的复芬斯勒度量的若干问题

Finsler几何是没有二次型限制的Riemann几何。具有特殊曲率性质的Finsler度量的研究，包括其构造和分类一直是Finsler几何的重要研究课题。酉不变的复Finsler度量是经典的复欧几里得度量、超球上的Bergmann度量和复射影空间上的Fubini-Study度量的推广，具有良好的对称性和可计算性。在本项目中我们将研究特殊的酉不变的复Finsler度量的构造和分类问题。主要内容包括以下三个方面：第一，研究光滑的、酉不变的弱Kähler-Finsler度量的构造及其分类问题；第二，研究复Einstein-Finsler度量的性质、构造和分类问题，尤其是强凸的、酉不变的复Einstein-Finsler度量和实Einstein-Finsler度量的关系。第三，研究强凸的、酉不变的、射影Ricci平坦的复Finsler度量的构造和分类问题。

Finsler geometry is just Riemann geometry without quadratic restrictions. The study of Finsler metrics with specific curvature properties, including its construction and classification has been an important topic in Finsler geometry. Unitary invariant complex Finsler metrics are generalizations of the classical complex Euclidean metric, the Bergmann metric in the unit ball and the Fubini-Study metric in complex projective space, and they enjoy good symmetry and above all computable. In this project, we shall study the constructions and classifications of special unitary invariant complex Finsler metrics. The main contents are as follows: Firstly,we shall study the constructions and classifications of smooth unitary invariant weakly Kähler-Finsler metrics; Secondly, we shall study the properties, constructions and classifications of complex Einstein-Finsler metrics, especially the relationship between the unitary invariant complex and real Einstein-Finsler metrics; Thirdly, we shall study the constructions and classifications of strongly convex unitary invariant projective Ricci flat complex Finsler metrics.