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

• 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
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• 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 / Neumann条件 / ラプラシアンの固有値 / ラプラシアン / 測度集中 / 曲率次元条件 / 測度の集中現象 / リッチ曲率

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.

Research Products

• [Journal Article] Estimates of eigenvalues of the Laplacian by a reduced number of subsets (2017)
  • Author(s): Kei Funano
  • Journal Title: Israel Journal of Mathematics Volume: 217 Issue: 1 Pages: 413-433
  • DOI: 10.1007/s11856-017-1453-7
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian (2016)
  • Author(s): Kei Funano
  • Journal Title: Analysis and Geometry in Metric Spaces Volume: 4 Issue: 1 Pages: 317-325
  • DOI: 10.1515/agms-2016-0015
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Estimates of eigenvalues of Laplacian by a reduced number of subsets (2016)
  • Author(s): Kei Funano
  • Journal Title: Israel Journal of Mathematics Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Concentration, Ricci curvature, and eigenvalues of Laplacian (2013)
  • Author(s): Kei Funano and Takashi Shioya
  • Journal Title: Geom. Funct. Anal. Volume: 23 Issue: 3 Pages: 888-936
  • DOI: 10.1007/s00039-013-0215-x
  • Peer Reviewed
• [Presentation] ハムサンドイッチとラプラシアン (2017)
  • Author(s): 船野敬
  • Organizer: リーマン幾何と幾何解析
  • Place of Presentation: 京都大学(京都府京都市)
  • Invited
• [Presentation] ハムサンドイッチとラプラシアン (2016)
  • Author(s): 船野敬
  • Organizer: 幾何と解析セミナー
  • Place of Presentation: 東北大学(宮城県仙台市)
  • Invited
• [Presentation] 巨視的スカラー曲率について (2016)
  • Author(s): 船野敬
  • Organizer: 微分トポロジーセミナー
  • Place of Presentation: 京都大学(京都府京都市)
  • Invited
• [Presentation] Estimates of ratios of consecutive eigenvalues of Laplacian via ham sandwich (2016)
  • Author(s): 船野敬
  • Organizer: young mathematicians seminar
  • Place of Presentation: Grenoble (France)
  • Int'l Joint Research
• [Presentation] Estimates of eigenvalues of Laplacian by a reduced number of subsets (2015)
  • Author(s): Kei Funano
  • Organizer: Geometry of Moduli Space of Low Dimensional Manifolds
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-12-15
  • Int'l Joint Research / Invited
• [Presentation] Estimates of eigenvalues of Laplacian by a reduced number of subsets (2015)
  • Author(s): Kei Funano
  • Organizer: Yau's student seminar
  • Place of Presentation: Harvard university
  • Year and Date: 2015-11-20
  • Int'l Joint Research / Invited
• [Presentation] Some upper bound estimate of eigenvalues of Laplacian on Alexandrov spaces of CD(0,\infty) (2014)
  • Author(s): 船野敬
  • Organizer: Seminar with E. Milman
  • Place of Presentation: 京都大学数学教室
  • Year and Date: 2014-09-30
  • Invited
• [Presentation] ラプラシアンの固有値の評価と分布 (2014)
  • Author(s): 船野敬
  • Organizer: Okayama Analysis and Probability Seminar
  • Place of Presentation: 岡山大学数学教室
  • Year and Date: 2014-06-09
  • Invited
• [Presentation] Eigenvalues of Laplacian and multi-way isoperimetric constants (2014)
  • Author(s): 船野敬
  • Organizer: 微分トポロジーセミナー
  • Place of Presentation: 京都大学数学教室
  • Year and Date: 2014-04-22
  • Invited
• [Presentation] リーマン多様体上のラプラシアンの固有値と多重等周定数について (2014)
  • Author(s): 船野敬
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学
  • Invited
• [Presentation] Eigenvalues of Laplacian and Multi-way isoperimetric constants on Riemannian manifolds I (2013)
  • Author(s): 船野敬
  • Organizer: Oberseminar Geometric Analysis
  • Place of Presentation: Bielefeld大学
  • Invited
• [Presentation] Eigenvalues of Laplacian and Multi-way isoperimetric constants on Riemannian manifolds II (2013)
  • Author(s): 船野敬
  • Organizer: Oberseminar Geometric Analysis
  • Place of Presentation: Bielefeld大学
  • Invited
• [Presentation] Eigenvalues of Laplacian and Multi-way isoperimetric constants on Riemannian manifolds III (2013)
  • Author(s): 船野敬
  • Organizer: Oberseminar Geometric Analysis
  • Place of Presentation: Bielefeld大学
  • Invited
• [Presentation] Eigenvalues of Laplacian and Multi-way isoperimetric constants on Riemannian manifolds (2013)
  • Author(s): 船野敬
  • Organizer: The Ninth Geometry Conference for the friendship between Japan and China
  • Place of Presentation: 登別グランドホテル
  • Invited
• [Presentation] Eigenvalues of Laplacian and Multi-way isoperimetric constants on Riemannian manifolds (2013)
  • Author(s): 船野敬
  • Organizer: Seminaire d'Algebres d'Operateurs
  • Place of Presentation: パリ第７大学
  • Invited
• [Presentation] Eigenvalues of Laplacian on Riemannian manifolds of nonnegative Ricci curvature (2013)
  • Author(s): 船野敬
  • Organizer: Mathematical Seminar
  • Place of Presentation: Institut des Hautes Etudes Scientifiques
  • Invited
• [Presentation] Eigenvalues of Laplacian and Multi-way isoperimetric constants on Riemannian manifolds (2013)
  • Author(s): 船野敬
  • Organizer: Working seminar "Courbure, transport optimal et probabilites"
  • Place of Presentation: Institut Henri Poincare
  • Invited
• [Book] 数理科学 2016年12月号 No.642 特集：「高次元世界の数学」 － なぜ次元を上げるのか － (2016)
  • Author(s): 船野敬
  • Total Pages: 7
  • Publisher: サイエンス社
• [Remarks] Kei Funano's homepage
  • URL: https://sites.google.com/site/keifunanoshomepage/