Study of metric measure spaces with curvature-dimension conditions and its applications to Riemannian geometry

• Project/Area Number: 18K13412
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kumamoto University
• Principal Investigator: Kitabeppu Yu 熊本大学, 大学院先端科学研究部(理), 准教授 (50728350)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 熱流 / RCD 空間 / 曲率次元条件 / 非線形ラプラシアン / 最大直径定理 / Gromov-Hausdorff 収束 / Ricci 曲率 / リーマン的曲率次元条件 / Wasserstein 距離 / Symplectic toric 多様体 / リーマン幾何学

Outline of Final Research Achievements

It is known that we can use much techniques of calculus on RCD spaces, which can be applied to wide class including collapsed manifolds.
Roughly, we have the following two results: One is the Zhong-Yang type rigidity theorem for RCD spaces, and another is giving a definition of Ricci curvature on hypergraphs associated with the heat flow.

Academic Significance and Societal Importance of the Research Achievements

ハイパーグラフはネットワークのモデルなどとしてよく使われており、特に近年ハイパーグラフ上の熱の伝わりかたに関する研究もよくされている。ハイパーグラフ上に熱流、正確にいえば非線形ラプラシアンから定まるレゾルベント、を用いた曲率の定義を与えたことで、今後ハイパーグラフ上の現実的な問題に関しても応用があると期待される。

Research Products

• [Journal Article] The rigidity of sharp spectral gap in non-negatively curved spaces (2023)
  • Author(s): Christian Ketterer, Yu Kitabeppu, Sajjard Lakzian
  • Journal Title: Nonlinear Analysis Volume: 228 Pages: 113202-113202
  • DOI: 10.1016/j.na.2022.113202
  • Peer Reviewed / Open Access
• [Journal Article] Cheng's maximal diameter theorem for hypergraphs (2023)
  • Author(s): Yu Kitabeppu, Erina Matsumoto
  • Journal Title: Tohoku Math. J. Volume: 75 Issue: 1 Pages: 119-130
  • DOI: 10.2748/tmj.20211202
  • Peer Reviewed / Open Access
• [Journal Article] Cheng's maximal diameter theorem for hypergraphs (2022)
  • Author(s): Yu Kitabeppu, Erina Matsumoto
  • Journal Title: Tohoku Mathematical Journal Volume: -
  • Peer Reviewed
• [Journal Article] A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces (2018)
  • Author(s): Kitabeppu Yu
  • Journal Title: Potential Analysis Volume: 印刷中 Issue: 2 Pages: 179-196
  • DOI: 10.1007/s11118-018-9708-4
  • Peer Reviewed
• [Presentation] ハイパーグラフ上のリッチ曲率 (2022)
  • Author(s): 北別府悠
  • Organizer: 第69回幾何学シンポジウム
  • Invited
• [Presentation] Coarse Ricci curvature on hypergraphs (2022)
  • Author(s): Yu Kitabeppu
  • Organizer: 第７回日中幾何学研究集会
  • Int'l Joint Research / Invited
• [Presentation] Zhong-Yang 型スペクトルギャップの剛性定理 (2022)
  • Author(s): 北別府悠
  • Organizer: 確率論と幾何学
  • Invited
• [Presentation] Rigidity theorem of Zhong-Yang type spectral gap for RCD(0,N) spaces (2021)
  • Author(s): 北別府悠
  • Organizer: Elliptic and Parabolic Zoom Seminar
  • Invited
• [Presentation] ハイパーグラフ上のリッチ曲率 (2021)
  • Author(s): 北別府悠
  • Organizer: リーマン幾何と幾何解析
  • Invited
• [Presentation] Delzant多面体の収束と対応するトーリック多様体の収束の関係について (2020)
  • Author(s): 北別府悠
  • Organizer: 第67回 幾何学シンポジウム
  • Invited
• [Presentation] ハイパーグラフ上の Ricci 曲率について (2020)
  • Author(s): 北別府悠
  • Organizer: 淡路島幾何学研究集会2020
  • Invited
• [Presentation] Lin-Lu-Yau の coarse Ricci 曲率について (2019)
  • Author(s): 北別府悠
  • Organizer: Study meeting on graphs and curvature
  • Invited
• [Presentation] Convergences of symplectic toric manifolds and the corresponding Delzant polytopes (2019)
  • Author(s): Yu Kitabeppu
  • Organizer: Geometry and Probability
  • Int'l Joint Research / Invited
• [Presentation] Delzant 多面体の収束と対応する多様体の収束について (2019)
  • Author(s): 北別府悠
  • Organizer: 淡路島幾何学研究集会2019
  • Invited
• [Presentation] 閉凸集合の間の新しい距離について (2018)
  • Author(s): 北別府悠
  • Organizer: 福岡大学微分幾何学研究集会2018
  • Invited
• [Presentation] The lower semi-continuity of the dimension of RCD spaces under the Gromov-Hausdorff convergence (2018)
  • Author(s): Yu Kitabeppu
  • Organizer: 4-th China-Japan Geometry conference
  • Int'l Joint Research / Invited