Rigidity of non-isometric actions of discrete groups and non-linear spectral gap

• Project/Area Number: 17H02840
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: Nayatani Shin 名古屋大学, 多元数理科学研究科, 教授 (70222180)
• Co-Investigator(Kenkyū-buntansha): 井関 裕靖 慶應義塾大学, 理工学部(矢上), 教授 (90244409)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥17,290,000 (Direct Cost: ¥13,300,000、Indirect Cost: ¥3,990,000); Fiscal Year 2021: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2019: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2018: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2017: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
• Keywords: スペクトルギャップ最大化 / 最適埋め込み / 離散群の剛性 / 球面内の極小曲面 / ラプラシアン第1固有値最大化 / 多様体の最適埋め込み / アフィン等長作用 / 離散群の超剛性 / 非等長的作用 / (非)線形スペクトルギャップ / グラフの埋め込み不変量 / 極小曲面 / グラフの最適埋め込み / 球面内の極小閉曲面 / ランダム群 / アフィン作用 / 非線形スペクトルギャップ / スペクトルギャップ / 高次元多面体

Outline of Final Research Achievements

We studied the optimization problems concerning embeddings into Euclidean spaces and the linear spectral gap of a finite graph and were able to find optimal solutions for a distance-regular graph. We studied the problem of finding an edge-length function maximizing the linear spectral gap and proved a Nadirashvili type theorem. We studied a new optimization problem concerning embeddings and the linear spectral gap of a manifold. We presented some examples where the problems can be solved and proved a Nadirashvili type theorem. We showed that a discrete equivariant harmonic map form a finitely generated group equipped with a random walk induces a boundary map under appropriate assumptions.

Academic Significance and Societal Importance of the Research Achievements

本研究は, ラプラシアン第1固有値のように, それ自身変分問題の最適値であるものをリーマン計量をすべて動かしてさらに最大化するという, 高次の変分問題を扱っており, 数学研究の新たな発展に関わるものと考えている. 新たにNash等長埋め込みと関連する双対最適化問題を設定したことも意義があろう. 離散と連続にまたがる研究であり, 材料科学への示唆も期待できよう.

Research Products

• [Journal Article] Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian (2019)
  • Author(s): Shin Nayatani, Toshihiro Shoda
  • Journal Title: Comptes Rendus Mathematique, Academie des Sciences, Paris Volume: 357 Pages: 84-98
  • Peer Reviewed / Open Access
• [Journal Article] Fixed-point property for affine actions on a Hilbert space (2017)
  • Author(s): Shin Nayatani
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B66 Pages: 115-131
  • Peer Reviewed
• [Presentation] First-eigenvalue maximization and embedding optimization (2021)
  • Author(s): 納谷信
  • Organizer: The 3rd Japan-Taiwan Joint Conference on Differential Geometry
  • Int'l Joint Research / Invited
• [Presentation] First-eigenvalue maximization and embedding optimization (2021)
  • Author(s): 納谷信
  • Organizer: 第６回日中幾何学研究集会
• [Presentation] 有限グラフの埋め込み不変量の最小化と第1固有値の最大化 (2020)
  • Author(s): 五明工, 納谷信
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Metrics maximizing the first eigenvalue of the Laplacian on a closed surface (2019)
  • Author(s): 納谷 信
  • Organizer: Mathematics Colloquium, University of Warwick
  • Invited
• [Presentation] ラプラシアンの第1固有値を最大化する閉曲面上の計量について1, 2 (2019)
  • Author(s): 納谷 信
  • Organizer: Workshop on geometry of networks and related topics in Kagoshima
  • Invited
• [Presentation] Riemannian metrics maximizing the first eigenvalue of the Laplacian on a closed surface (2019)
  • Author(s): 納谷 信
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Invited
• [Presentation] Riemannian metrics maximizing the first eigenvalue of the Laplacian on a closed surface (2019)
  • Author(s): 納谷 信
  • Organizer: The first Geometry Conference for Friendship of Japan and Germany
  • Int'l Joint Research / Invited
• [Presentation] 有限グラフの第1固有値の最大化と埋め込み不変量の最小化 (2019)
  • Author(s): 納谷 信
  • Organizer: 福岡大学微分幾何研究集会
  • Invited
• [Presentation] Riemannian metrics maximizing the first eigenvalue of the Laplacian on a closed surface (2019)
  • Author(s): 納谷 信
  • Organizer: 研究集会 Geometry and Analysis
  • Int'l Joint Research / Invited
• [Presentation] Metrics maximizing the first eigenvalue of the Laplacian on a closed surface and extra eigenfunction (Mini-course) (2019)
  • Author(s): Shin Nayatani
  • Organizer: UK-Japan Winter School "Variational problems in geometry and mathematical physics"
  • Int'l Joint Research / Invited
• [Presentation] 高次元多面体と極小曲面 (2019)
  • Author(s): Shin Nayatani
  • Organizer: 高次元多面体勉強会
• [Presentation] ラプラシアンの第1固有値を最大化する閉曲面上の計量について (2018)
  • Author(s): Shin Nayatani
  • Organizer: 福島幾何学研究集会2018
  • Invited
• [Presentation] ラプラシアンの第1固有値の最大化と極小曲面(企画特別講演) (2018)
  • Author(s): Shin Nayatani
  • Organizer: 2018年度秋季総合分科会
• [Presentation] Bolza曲面上の三つの計量 (2018)
  • Author(s): Shin Nayatani
  • Organizer: 研究集会「Geometry of Riemann surfaces and related topics」
  • Invited
• [Presentation] ラプラシアンの第１固有値を最大化する閉曲面上の計量について (2018)
  • Author(s): 納谷 信
  • Organizer: 研究集会「リーマン幾何と幾何解析」
  • Invited
• [Presentation] ラプラシアンの第１固有値を最大化する種数２閉曲面上の計量 (2018)
  • Author(s): 庄田 敏宏、納谷 信
  • Organizer: 日本数学会2018年度年会
• [Presentation] Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian (2017)
  • Author(s): Shin nayatani
  • Organizer: 第３回日中幾何学研究集会(The 3rd Japan-China Geometry Conference)
  • Int'l Joint Research / Invited
• [Presentation] ラプラシアンの第１固有値を最大化する閉曲面上の計量について (2017)
  • Author(s): 納谷 信
  • Organizer: 福岡大学微分幾何セミナー
• [Presentation] Hersch-Yang-Yauの不等式と閉曲面上の第１固有値の最大化について (2017)
  • Author(s): 納谷 信
  • Organizer: Workshop 「Geometric Analysis in Geometry and Topology 2017」
  • Invited
• [Funded Workshop] RIMS Review Seminar, Symmetry and Stability in Differential Geometry of Surfaces (2022)
• [Funded Workshop] Rigidity School Nagoya 2018 (2018)
• [Funded Workshop] Rigidity School Nagoya 2017 (2017)