流形上的谱分析及曲率流在其上的应用

This project devotes to the study of spectral analysis on Riemannian manifolds and investigates the application of curvature flows in the spectral analysis on manifolds. In order to carry out our project well, we issue several specified problems from the aspects “eigenvalue comparison theorems”, “eigenvalue estimation”, “the study of the existence of the ground state of quantum layers”, “monotonicity of eigenvalues under the specified curvature flow and its application, the tool of curvature flows and isoperimetric inequalities”, and meanwhile we also give corresponding feasible research approaches to those problems. We believe that in the process of trying to solve those specified problems, our capacity of mastering the analytic techniques on manifolds and the tool of Partial Differential Equations will be improved.. “Spectral analysis on manifolds” and “curvature flows” are hot topics in Geometric Analysis. Many mathematicians have been working on these two fields and many symbolic achievements have been obtained, such as Cheng’s eigenvalue comparison theorems [S.-Y. Cheng, Eigenvalue comparison theorems and its geometric application, Math. Z. 143 (1975) 289–297; S.-Y. Cheng, Eigenfunctions and eigenvalues of the Laplacian, Am. Math. Soc. Proc. Symp. Pure Math. 27 (Part II) (1975) 185–193], the complete proof of the fundamental gap conjecture for a class of Schrödinger operators [B. Andrews and J. Clutterbuck, Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 24 (3) (2011) 899-916], the proof of 3-dimensional Poincaré conjecture given by G. Perelman, and so on.. Because of this fact, I propose this research project which has relations with the topics “spectral analysis on manifolds” and “curvature flows”, and especially, several specified problems within the scope of my academic ability have been issued to make sure that the project runs fluently. Hence, this project is reasonable, and is also meaningful.

本项目致力于黎曼流形上的谱分析研究以及探讨曲率流在谱分析里的应用。为了更好地开展研究，我们从“特征值比较定理”、“特征值估计”、“量子层的基态的存在性研究”、“特征值在曲率流下的单调性及其应用、曲率流工具与等周不等式”这几个方面入手提出了若干具体的问题，并且同时也可给出了针对它们的可行的研究方案。我们相信在努力解决这些具体问题的过程中，将会进一步地提升我们对于流形上的分析技巧以及偏微分方程工具的掌握。. “流形上的谱分析”与“曲率流”是几何分析的热门研究领域，有不少学者致力于这些方面的研究，取得了很多标志性的研究成果，比如：Cheng-特征值比较定理、基本间隔猜想及3-维庞加莱猜想的证明等等。. 正是基于这一事实，本人提出这一同“流形上的谱分析”与“曲率流”有着联系的研究项目，尤其是给出了若干能力范围之内的具体问题以保证项目的顺利进行。因此，本项目是合理的，很有意义。

本项目致力于“流形上的谱分析”并且探讨“曲率流”理论在谱分析里的应用，它们均是当下微分几何研究的热点专题。为了使研究更具有针对性，依据研究计划，我们的研究围绕若干具体的问题展开，经过三年的潜心研究，在国家自然科学基金委的资助下，取得了丰硕的研究成果，大大地超过了预期目标（原拟定完成3-5篇SCI论文，但截至目前，经过统计，本人及主要参与者共计在国内外学术期刊上发表已署名本项目资助的学术论文19篇，其中被SCI收录16篇,部分结果发表在《J. Differential Equat.》、《Stoch. Proc. Appl.》、《J. Geom. Anal.》、《Ann. Glob. Anal. Geom.》、《Commun. Pure Appl. Anal.》、《Arch. Math.》等重要国际学术期刊上。此外，另有3篇学术论文被接收，在科学出版社出版英文学术著作《Eigenvalue Problems on Manifolds》一部），高质、高效地完成了既定的研究任务。