关于 Finsler 流形上调和映射与 Laplacian 的若干问题研究

Harmonic maps and Laplacian are very important issuses in globle differential geometry and geometric analysis, which have been studied intensively and comprehensively in Riemannian geometry, and the relatively perfect theory systems have been formed. However, in Finsler case, the tedious calculations and complicated nonlinear problems have become the main obstacles in the course of studies. This project aims to study the corresponding topics in Finsler geometry. Specifically, we are to study geometric properties, including the existence and the stability, of harmonic maps and minimal immersions into Finsler manifolds. We will generalize further the Bernstein type theorems, the Liouville type theorems and other rigidity theorems. We will also consider some problems on the first eigenvalue of Finsler Laplacian and Finsler p-Laplacian. We shall not only give some estimates and comparison theorems, but also build some rigidity theorems. Moreover, We will investigate the curvature properties and metric properties of Finsler manifolds and submanifolds with special metrics, such as Minkowski spaces, Randers spaces and (α,β)-metric spaces. The research of this project can be viewed as a generalization and development of the corresponding topics in the Riemannian case.The main purpose is to enrich and improve theoretical results in Finsler geometric analysis and explore new ideas and methods.

调和映射和 Laplacian是整体微分几何及几何分析的重要研究课题。在黎曼几何中，这些问题得到了广泛而深入的研究，形成了较为完善的理论体系。然而在Finsler 情形下，相关的计算更为繁琐、非线性问题更为复杂，成为研究的主要障碍。 本项主要研究Finsler几何中的相应问题，内容包括：研究到Finsler流形的调和映射和极小浸入的几何性质，如存在性和稳定性问题，进一步推广Bernstein 型定理、Liouville型定理和其他刚性定理；考虑Finsler (p-)Laplacian的第一特征值问题，给出各种估计及比较定理，建立相应的刚性定理；研究Minkowski空间、Randers空间、(α,β)空间及其子流形的曲率性质和度量性质。 本项目的研究是黎曼情形下相关问题的推广与延伸，旨在充实和完善整体Finsler几何的理论成果，探索新的方法与思路.

调和映射和 Laplacian是整体微分几何及几何分析的重要研究课题。在黎曼几何中，这些.问题得到了广泛而深入的研究，形成了较为完善的理论体系。然而在Finsler 情形下，先关研究结果还比较少。 本项主要研究Finsler.几何中Laplacian的相关问题，内容包括：.研究Finsler (p-)Laplacian的第一特征值问题，给出了各种上下界估计和比较定理，建立了相.应的刚性定理；.研究等参超曲面的几何性质和存在性，给出了若干分类结果；.研究Minkowski空间、Randers空间、(α,β)空间的曲率性质.和度量性质，证明了爱因斯坦度量的的存在性问题，并给出了共形平坦和对偶平坦芬斯勒度量一些分类结果。 .本项目的研究是黎曼情形下相关问题的推广与延伸，旨在充实和完善整体F.insler几何的理论成果，探索新的方法与思路。在这两个方面，本项目均取得了实质性进展。