澳门威尼斯人

一、基本信息

题目：几何分析短课程：特征值的万有不等式

报告人：罗勇 教授

报告人单位：重庆理工大学

报告人简介：罗勇，重庆理工大学数学科学研究中心教授，巴渝学者特聘教授。2007年6月本科毕业于武汉大学，2011年1月硕士毕业于中国科学院数学与系统科学研究院，导师为王友德研究员，2013年12月博士毕业于德国弗莱堡大学，导师为王国芳教授。2014年1月至2020年10月在武汉大学工作，历任师资博士后、讲师、副教授。2015年7月至2016年6月在德国莱比锡马克斯普朗克应用数学所做博士后，合作导师为德国自然科学院院士Jurgen Jost教授。2020年11月起在重庆理工大学工作。研究方向为微分几何与几何分析，近年来主要关注和研究子流形几何、曲率流的奇点分析、特征值问题等。

课程安排：

课程简介：Inequalities for eigenvalues of certain elliptic operator with natural boundary conditions defined on bounded domain on a complete Riemannian manifold are called universal inequalities, if they require no hypotheses on the geometric quantities of the domain (other than its dimension). In the last seventy years, universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem attracted much attention and a lot of works on this subject have been done.

In this mini-course, we will discuss Topics which include:

1. Yang’s inequality for eigenvalues of the Dirichlet Laplacian and its various generalizations.

2. Cheng and Yang’s recursion formula and upper bound estimates of eigenvalues.

3. Our recent work on universal inequalities.

二、相关文献

[1] D. G. Chen , Q. M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator. J Math Soc Japan, 2008, 60, no.2: 325–339.

[2] Q. M. Cheng, Eigenvalue Estimates for Eigenvalues and Applications. Adv Lect Math (ALM), International Press, Somerville, MA, 2017, 60: 37–52.

[3] Q. M. Cheng and H. C. Yang, Bounds on eigenvalues of Dirichlet Laplacian. Math Ann, 2007, 337: 159–175.

[4] Q. M. Cheng and H. C. Yang, Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc. 358 (2006), no.6, 2625–2635.

[5] Q. M. Cheng, T. Ichikawa, S. Mametsuka, Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds, J. Math. Soc. Japan. 62 (2010), 673–686.

[6] Y. Luo and X. J. Zheng, Inequalities for eigenvalues of Laplacian and biharmonic operators on submanifolds. J. Math. Study, 2025, to appear.

[7] Y. Luo and X. J. Zheng, Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds, Acta Math. Sci. Ser. B (Engl. Ed.), 2025, to appear.

[8] A. Soufi, E. M. Harrel II and S. Ilias, Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds, Trans. Amer. Math. Soc. 361 (2009), no.5, 2337–2350.

[9] Q. L. Wang and C. Y. Xia, Inequalities for eigenvalues of a clamped plate problem, Calc. Var. and Partial Differetial Equations 40 (2011), 273–289.

[10] ] H. C. Yang, An estimate of the difference between consecutive eigenvalues, Preprint IC/91/60 of ICTP (1991), Trieste.