Applications of the concentration of measure phenomenon to analysis and geometry of Laplacian

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800042
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tohoku University (2016); Kyoto University (2013-2015)
• Principal Investigator: Funano Kei 東北大学, 情報科学研究科, 准教授 (40614144)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: Laplacianの固有値 / Ricci曲率 / 凸体 / 普遍不等式 / ham sandwichの定理 / 最適輸送

Outline of Final Research Achievements

I obtained some upper bound estimates of eigenvalues of the Laplacian on closed Riemannian manifolds of nonnegative Ricci curvature. These estimates state that one can estimate eigenvalues in terms of infomation of finite number of subsets of the manifold. The method I used in the proof is the theory of optimal transportation.
I also studied domain monotonicity/reverse domain monotonicity for Neumann eigenvalues of the Laplacian on convex domains in a Euclidean space. Furthermore I got nontrivial universal inequalities among eigenvalues of the Laplacian. In the proof I used the ham sandwich theorem coming from algebraic topology. These studies are valuable.

Free Research Field

スペクトル幾何学