芬斯勒几何中若干重要曲率性质及相关问题研究

This project will focus on the research of flag curvature, Ricci curvature and scalar curvature in Finsler geometry as well as some related important and profound topics. We will devote ourselves to developing the geometry of Finsler spaces of scalar flag curvature and will thoroughly explore the geometric and analytical properties of Ricci curvature and scalar curvature. We will deeply reveal the relationships between the curvature pinching conditions of Finsler manifolds and the topology of manifolds. The research topics involve Finsler Ricci curvature flow and Finsler Ricci soliton geometry, the Yamabe problem in Finsler geometry and the related positive mass conjecture. The project will also make an in-depth study of the sphere theorem with flag curvature pinching condition, the structure of Finsler manifolds with non-negative flag curvature as well as the generalizations of Cheeger finiteness theorem and Cheeger-Gromov's collapsing theorem in Finsler Geometry. This project is driven by some famous problems in Finsler geometry and is led by some key problems that need eagerly to be solved in the development of Finsler geometry. It not only highlights the research on the geometric structures and global properties of Finsler manifolds, but also strengthens the research on the global analysis and topology on Finsler manifolds. The research topics in this project are new and original and are of great significance to promote the development of Finsler geometry at home and abroad.

本项目拟围绕芬斯勒几何中的旗曲率、Ricci曲率和数量曲率以及相关的若干重要而深刻的问题展开研究，致力发展具有标量旗曲率的芬斯勒空间的几何，深入探讨Ricci曲率和数量曲率的几何与分析性质，深入揭示芬斯勒流形的曲率拼挤条件与流形拓扑的关系。本项目研究内容涉及到芬斯勒Ricci曲率流以及相应的Ricci soliton几何、芬斯勒几何中的Yamabe问题及与之密切相关的正质量猜想，并将深入研究旗曲率拼挤条件下的球面定理、非负旗曲率芬斯勒流形的结构以及Cheeger有限性定理和Cheeger-Gromov的坍缩定理在芬斯勒几何中的推广。本项目以芬斯勒几何中的若干著名问题为驱动，以芬斯勒几何发展过程中亟待解决的若干关键问题为引领，既突出了关于芬斯勒流形几何结构与整体性质的研究，又强化了芬斯勒流形上整体分析与拓扑方面的研究，研究内容新颖，对推进国内外芬斯勒几何的发展具有非常重要的意义。

本项目对具有标量旗曲率的芬斯勒度量开展了继续深入的研究，进一步刻画和揭示了这类芬斯勒度量的几何与分析性质。继续深入探讨了芬斯勒几何中的Ricci曲率性质，围绕Einstein芬斯勒度量的存在性问题开展了一系列深入研究。我们深入研究了芬斯勒几何中的数量曲率对芬斯勒空间结构的影响，深入探讨和揭示了数量曲率的共形性质；深入揭示了加权Ricci曲率有界的芬斯勒流形的整体几何与分析性质。我们还系统研究了指标理论中的超联络理论和几何局部化方法在芬斯勒流形上的Gauss-Bonnet-陈定理等方面的应用，深刻揭示了复芬斯勒度量的正交Ricci曲率对复芬斯勒空间结构的影响。作为芬斯勒几何中的一项重要的基础性研究工作，我们继续深入研究了芬斯勒流形上共形向量场的性质，利用共形向量场刻画和揭示了芬斯勒空间的几何结构；继续深入研究了若干非黎曼几何量的性质，深入探讨了芬斯勒几何中著名的“独角兽”问题。四年来，在基金的支持下，本项目开展了一系列富有成效的学术交流活动；已发表论文32篇；另有4篇论文被录用待发表；另完成论文7篇。本项目的研究成果已在国内外同行中产生了广泛而重要影响。