The latest frontier study on geometry and theory of eigenvalues

2016 Fiscal Year Final Research Report

• Project/Area Number: 24340013
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Fukuoka University
• Principal Investigator: Cheng Qing-Ming 福岡大学, 理学部, 教授 (50274577)
• Co-Investigator(Kenkyū-buntansha): 山田 光太郎 東京工業大学, 理学院, 教授 (10221657) ; 塩谷 隆 東北大学, 理学(系)研究科(研究院), 教授 (90235507)
• Co-Investigator(Renkei-kenkyūsha): Futaki Akito 東京大学, 数理科学研究科, 教授 (90143247) ; Koiso Miyuki 九州大学, 数理学研究院, 教授 (10178189) ; Rossman Wayne 神戸大学, 理学研究科, 教授 (50284485)
• Research Collaborator: Li Haizhong 清華大学, 教授 ; Wei Guoxin 華南師範大学, 教授 ; Yang Hongcang 中国科学院, 教授
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: eigenvalues of Laplacian / Riemannian manifold / minimal immersion / Alenxandrov space / Singularity / maximum principle / mean curvature flow / self-shrinker

Outline of Final Research Achievements

By making use of Cheng-Yang recursion formula, we give optimal estimates for lower bounds of eigenvalues of Laplacian on a bounded domain in complete Riemannian manifolds. Our method is original. According to this result, a difficult problem proposed by I. Chavel is solved. Furthermore, we find an obstruction on minimal immersions from complete Riemannian manifolds into Euclidean spaces in the view of eigenvalues of Laplacian. Geometry of fronts with singularities has been studied. Gauss-Bonnet theorem on fronts is proved. By improving the generalized maximum principle of Omori-Yau, important results on classification of complete self-shrinkers of the mean curvature flow are obtained. Eigenvalues of Laplacian on compact Alexandrov spaces are studied and important progresses are obtained.

Free Research Field

微分幾何学