Studies of geometry of singularities and the eigenvalues of the Hodge-Laplacian

• Project/Area Number: 16K05117
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Takahashi Junya 東北大学, 情報科学研究科, 助教 (10361156)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: ホッジ・ラプラシアン / ラフ・ラプラシアン / 微分形式 / 固有値 / リーマン計量の変形 / 特異点 / L2 ストークス定理 / L2 調和形式 / 断面曲率 / 体積一定 / リーマン多様体の崩壊 / 固有値の収束 / 球面 / 非負断面曲率 / リーマン沈め込み / ベッチ数 / ホッジ理論 / L^2 調和形式 / L^2 ストークス定理 / 消滅定理 / ラプラシアン / 小さい固有値 / エッジ特異点 / 自己共役性 / 幾何学 / スペクトル

Outline of Final Research Achievements

We study to reveal the geometric information of the eigenvalues of the Hodge-Lalacian on closed Riemannian manifolds, and then obtained the following results:
On any m-dimensional closed manifold, for a given degree p with 1≦p≦m-1 and a natural number k, we construct a family of Riemannian metrics with fixed volume such that the k-th eigenvalues of the Hodge-Laplacian and the rough Laplacian acting on p-forms converge to 0. In particular, in the case of m-dimensional spheres, we can choose these Riemannian metrics as thoes of non-negative sectional curvature.
This is a joint work with Colette Ann'e (Universit'e de Nantes in France).

Academic Significance and Societal Importance of the Research Achievements

コンパクト Riemann 多様体上の Hodge-Laplacian の固有値の持つ幾何学的情報の解明は大変重要な問題である．しかし，未だその全容の解明には至っていない．解明には様々な困難があるが，まずは具体例を多く作ることが重要である．特に，幾何学的状況がつかみ易い例は重宝する．
今回，そのような具体例を構成出来たので，応用や今後の研究の方向性を与えることが出来たと考えている．

Research Products

• [Int'l Joint Research] ナント大学(フランス)
• [Int'l Joint Research] Universit'e de Nantes(フランス)
• [Int'l Joint Research] ナント大学(フランス)
• [Int'l Joint Research] Nantes 大学(フランス)
• [Int'l Joint Research] Jena 大学(ドイツ)
• [Journal Article] Small eigenvalues of the rough and Hodge Laplacians under fixed volume (2022)
  • Author(s): Colette Ann'e and Junya Takahashi
  • Journal Title: Annales de la Facult'e des Sciences de Toulouse Volume: -
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] L^2-harmonic forms on incomplete Riemannian manifolds with positive Ricci curvature (2018)
  • Author(s): Junya Takahashi
  • Journal Title: Mathematics Volume: 6 Issue: 5 Pages: 1-11
  • DOI: 10.3390/math6050075
  • Peer Reviewed / Open Access
• [Journal Article] $L^2$-harmonic forms on incomplete Riemannian manifolds with positive Ricci curvature (2018)
  • Author(s): Junya Takahashi
  • Journal Title: Mathematics, Special Issue, Differential Geometry Volume: 印刷中
  • Peer Reviewed / Open Access
• [Presentation] 体積一定下での Hodge-Laplacian と rough Laplacian の小さい固有値 (2022)
  • Author(s): 高橋淳也, Colette Ann'e
  • Organizer: 日本数学会幾何学分科会
• [Presentation] Partial collapsing and the spectrum of the Hodge-Laplacian (2016)
  • Author(s): 高橋淳也
  • Organizer: 多様体上の微分方程式
  • Place of Presentation: 金沢大学 サテライト・プラザ（石川県金沢市）
  • Year and Date: 2016-11-19
  • Invited