芬斯勒几何中某些整体分析与拓扑问题的研究

本项目主要研究芬斯勒流形上整体几何、分析和拓扑及其内在联系。主要内容有:(1)继续深化芬斯勒流形上特征值问题和热流等方面研究,包括不可反芬斯勒流形上拉普拉斯算子特征谱的存在性和完备芬斯勒流形上热方程非负解唯一性与其正解估计及应用等;(2)实(复)芬斯勒流形上曲率和拓扑关系的研究，包括建立芬斯勒几何中Gromov-Meyer型定理、芬斯勒流形的有限拓扑型和具有负全纯双截面曲率的完备单连通kahler芬斯勒流形刚性等;(3)实(复)芬斯勒流形上典则度量的几何分析及其存在性研究,特别关注Kahler芬斯勒流形上Kahler爱因斯坦芬斯勒度量的几何、分析性质和存在性等问题.通过本项目的研究,深化和完善芬斯勒几何中已有的研究成果,拓展芬斯勒几何研究内容和加深对芬斯勒流形上整体分析与拓扑的理解.本项目涉及到(微分、代数)几何、复分析、偏微分方程与拓扑等多门学科,是非常有意义的研究课题.

This project focuses on the global geometry, analysis and topology on Finsler manifolds and the inner relationships among them. The main contents are as follows. (1) Continue to deepen the research on the eigenvalue problem and heat flow etc. on a Finsler manifold, including the existence of the eigenvalue spectrum for the Laplacian on a nonreversible Finsler manifold, the uniqueness for the nonnegative solutions and the estimate of the positive solutions for the heat flow on a complete Finsler manifold and their applications etc.. (2) Research on the relationships between the curvature and topology on Finsler manifolds, including establishing the Gromov-Meyer theorem in Finsler geometry, the finite topology type of Finsler manifolds and the rigidity of complete simply connected Kahler Finsler manifolds with negative holomorphic bisectional curvature etc.. (3) Research on the geometric analysis and existence of the canonical metrics on real (complex) Finsler manifolds. In particular, we shall focus on the geometric or analytic properties and the existence of the Kahler Einstein Finsler metrics. By means of this project, we will deepen and improve the known results in Finsler geometry, extend the study contents of Finsler geometry and deepen the understandings of the global analysis and topology on Finsler manifolds. This project involves several subjects, such as, (differential, algebraic) geometry, complex analysis, partial differential equation and topology etc. and is a very meaningful research project.