Convergence・collapsing theory of manifolds, Ricci flows and the geometry and analysis of singular spaces

• Project/Area Number: 17204003
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: University of Tsukuba
• Principal Investigator: YAMAGUCHI Takao University of Tsukuba, 大学院・数理物質科学研究科, 教授 (00182444)
• Co-Investigator(Kenkyū-buntansha): FUKAYA Kenji 京都大学, 大学院・理学研究科, 教授 (30165261) ; KASUE Atsuhi 金沢大学, 大学院・自然科学研究科, 教授 (40152657) ; SHIOYA Takashi 東北大学, 大学院・理学研究科, 教授 (90235507) ; TODA Masahito お茶の水女子大学, 大学院・人間文化研究科, 教授 (80291566) ; ITOH Mitsuhiro 筑波大学, 大学院・数理物質科学研究科, 教授 (40015912) ; OTSU Yukio 九州大学, 大学院・数理学研究院, 准教授 (80233170) ; KAWAMURA Kazuhiro 筑波大学, 大学院・数理物質科学研究科, 准教授 (40204771) ; NAGANO Koichi 筑波大学, 大学院・数理物質科学研究科, 講師 (30333777) ; SATOH Hiroyasu 筑波大学, 大学院・数理物質科学研究科, 準研究員 (00375396)
• Project Period (FY): 2005 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥32,630,000 (Direct Cost: ¥25,100,000、Indirect Cost: ¥7,530,000); Fiscal Year 2008: ¥7,930,000 (Direct Cost: ¥6,100,000、Indirect Cost: ¥1,830,000); Fiscal Year 2007: ¥7,670,000 (Direct Cost: ¥5,900,000、Indirect Cost: ¥1,770,000); Fiscal Year 2006: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000); Fiscal Year 2005: ¥9,230,000 (Direct Cost: ¥7,100,000、Indirect Cost: ¥2,130,000)
• Keywords: リーマン多様体 / 崩壊 / アレクサンドロフ空間 / リッチ流 / 軌道体 / スペクトル逆問題 / 崩壊理論 / 退化 / 本質的被覆 / リッチ曲率 / ソウル定理 / 変分収束理論

Research Abstract

曲率が上に有界な2次元特異空間の局所構造を完全に決定し、ガウスボンネ型定理を得た(永野幸一、塩谷隆、B.Kleiner3氏との共同研究)。
断面曲率が下に直径が上に一様に有界な多様体或は特異空間に対して、本質的被覆なる新しい不変量を発見し、それが一様に有界であることを証明した。これによりグロモフ氏のベッチ数有界性定理の新しい見地からの別証明を得た。
ペレルマン氏のリッチ流に関する仕事を解明し、3次元多様体のリッチ流の特異時間における退化現象を解明した。
軌道体に対するゲルファンド・スペクトル逆問題を考察し、その局所スペクトラルデータから軌道体の等長同型構造が完全に決定されることを証明した(Y. Kurylev, M.Lassas 氏との共同研究)

Research Products

• [Journal Article] An upper curvature bound and singularities, Noncommutativity and Singularities (2009)
  • Author(s): T. Yamaguchi
  • Journal Title: Advanced Studies in Pure Mathematics 55 Pages: 161-171
  • Peer Reviewed
• [Journal Article] On the root closedness of continuous function algebras (2009)
  • Author(s): Kawamura, Kazuhiro; Miura, Takeshi
  • Journal Title: Topology Appl 156,no.3 Pages: 624-628
  • Peer Reviewed
• [Journal Article] On the root closedness of continuous function algebras, (2009)
  • Author(s): K. Kawamura; T. Miura
  • Journal Title: Top. Appl 156 Pages: 624-628
  • Peer Reviewed
• [Journal Article] Exact Lagrangian submanifolds in simply-connected cotangent bundles (2008)
  • Author(s): Fukaya, Kenji ; Seidel, Paul ; Smith, Ivan
  • Journal Title: Invent. Math 172,no. 1 Pages: 1-27
  • Peer Reviewed
• [Journal Article] Variational convergence over metric spaces,Trans (2008)
  • Author(s): Kuwae, Kazuhiro ; Shioya, Takashi
  • Journal Title: Amer. Math. Soc 360 Pages: 347-353
  • Peer Reviewed
• [Journal Article] A note on the Fisher information metric and heat kernels (2008)
  • Author(s): Itoh, Mitsuhiro ; Satoh, Hiroyasu ; Shishido, Yuichi
  • Journal Title: Int. J. Pure Appl. Math 46 Pages: 347-353
  • Peer Reviewed
• [Journal Article] The Serre duality theorem for a non-compact weighted CR manifold (2008)
  • Author(s): Itoh, Mitsuhiro ; Masamune, Jun ; Saotome, Takanari
  • Journal Title: Proc. Amer. Math. Soc 136no. 10 Pages: 3539-3548
  • NAID: 120007138743
  • Peer Reviewed
• [Journal Article] Fisher information metric and Poisson kernels (2008)
  • Author(s): Itoh, Mitsuhiro ; Shishido, Yuichi
  • Journal Title: Differential Geom. Appl 26no. 4 Pages: 347-356
  • NAID: 120007138689
  • Peer Reviewed
• [Journal Article] Variational convergence over metric spaces (2008)
  • Author(s): Kuwae, Kazuhiro; Shioya, Takashi
  • Journal Title: Trans. Amer. Math. Soc. 360 Pages: 35-75
  • NAID: 120003780146
  • Peer Reviewed
• [Journal Article] A note on the Fisher information metric and heat kernels. (2008)
  • Author(s): Itoh, Mitsuhiro; Satoh, Hiroyasu; Shishido, Yuichi
  • Journal Title: Int. J. Pure Appl. Math. 46 Pages: 347-353
  • Peer Reviewed
• [Journal Article] Variational convergence over metric spaces. (2008)
  • Author(s): Kuwae, Kazuhiro; Shioya, Takashi
  • Journal Title: Trans. Amer. Math. Soc. 360 Pages: 35-75
  • NAID: 120003780146
  • Peer Reviewed
• [Journal Article] The Serre duality for holomorphic vector bundles over strongly pseudo convex CR manifoldsProceedings of the Eleventh International Workshop on Differential Geometry (2007)
  • Author(s): Itoh, Mitsuhiro
  • Journal Title: Kyungpook Nat. Univ., Taegu, Pages: 17-24
• [Journal Article] 3-dim Seiberg-Witten theory and the Thurston's geometric structures.Proceedings of the Eleventh International Workshop on Differential Geometry (2007)
  • Author(s): Itoh, Mitsuhiro
  • Journal Title: Kyungpook Nat. Univ., Taegu Pages: 9-16
• [Journal Article] The Serre duality theorem for holomorphic vector bundles over a strongly pseudo-convex manifold (2007)
  • Author(s): Itoh, Mitsuhiro ; Saotome, Takanari
  • Journal Title: Tsukuba J. Math 31no. 1 Pages: 197-204
  • Peer Reviewed
• [Journal Article] Dirichlet finite harmonic functions and points at infinity of graphs and manifolds (2007)
  • Author(s): Hattori, Tae ; Kasue, Atsushi,
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci 83 Pages: 129-134
  • NAID: 40015669757
  • Peer Reviewed
• [Journal Article] On the existence of continuous (approximate) roots of algebraic equations (2007)
  • Author(s): Kawamura, Kazuhiro ; Miura, Takeshi
  • Journal Title: Topology Appl 154no. 2 Pages: 434-442
  • Peer Reviewed
• [Journal Article] Root closed function algebras on compacta of large dimension (2007)
  • Author(s): Brodskiy, N. ; Dydak, J. ; Karasev, A. ; Kawamura, K.
  • Journal Title: Proc. Amer. Math. Soc 135no. 2 Pages: 587-596
  • Peer Reviewed
• [Journal Article] Almost Hermitian structures on tangent bundles. Proceedings of the Eleventh International Workshop on Differential Geometry (2007)
  • Author(s): Satoh, Hiroyasu
  • Journal Title: Kyungpook Nat. Univ., Taegu Pages: 105-118
• [Journal Article] Dirichlet finite harmonic functions and points at infinity of graphs and manifolds. (2007)
  • Author(s): Hattori, Tae; Kasue, Atsushi
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. 83 Pages: 129-134
  • NAID: 40015669757
  • Peer Reviewed
• [Journal Article] The Serre duality theorem for holomorphic vector bundles over a strongly pseudo-convex manifold. (2007)
  • Author(s): Itoh, Mitsuhiro; Saotome, Takanari
  • Journal Title: Tsukuba J. Math. 31 Pages: 197-204
  • Peer Reviewed
• [Journal Article] 3-dim Seiberg-Witten Theory and thwe Thurston' s Geometric Structures (2007)
  • Author(s): Mitsuhiro Itoh
  • Journal Title: Proceedings of the Eleventh International Workshop on Diff. Geom Vol. 11 Pages: 9-16
• [Journal Article] Fixed point sets of parabolic isometries of CAT(0)-spaces, Comment (2006)
  • Author(s): Fujiwara, Koji ; Shioya, Takashi ;Koichi Nagano
  • Journal Title: Math. Helv 81 Pages: 1305-1335
  • Peer Reviewed
• [Journal Article] Floer homology in symplectic geometry and in mirror symmetry. International Congress of Mathematicians (2006)
  • Author(s): Oh, Yong-Geun ; Fukaya, Kenji
  • Journal Title: Eur. Math. Soc., Zurich Vol. II Pages: 879-905
  • Peer Reviewed
• [Journal Article] Application of Floer homology of Langrangian submanifolds to symplectic topology. [Application of Floer homology of Lagrangian submanifolds to symplectic topology] Morse theoretic methods in nonlinear analysis and in symplectic topology (2006)
  • Author(s): Fukaya, Kenji
  • Journal Title: NATO Sci. Ser. II Math. Phys. Chem.(Springer, Dordrecht) 217 Pages: 231-276
  • Peer Reviewed
• [Journal Article] Convergence of Riemannian manifolds and Laplace operators. II (2006)
  • Author(s): Kasue, Atsushi
  • Journal Title: Potential Anal 24 Pages: 137-194
  • Peer Reviewed
• [Journal Article] Metric Riemannian geometry (2006)
  • Author(s): Fukaya, Kenji
  • Journal Title: Handbook of differential geometry Vol. II Pages: 189-313
  • Peer Reviewed
• [Journal Article] Variational convergence of finite networks (2006)
  • Author(s): Kasue, Atsushi
  • Journal Title: Interdiscip. Inform. Sci 12no. 1 Pages: 57-70
  • NAID: 110004321831
  • Peer Reviewed
• [Journal Article] The dual Thurston norm and the geometry of closed 3-manifolds (2006)
  • Author(s): Itoh, Mitsuhiro ; Yamase, Takahisa
  • Journal Title: Osaka J. Math 43no.1 Pages: 121-129
  • Peer Reviewed
• [Journal Article] Fixed point sets of Parabolic isometries of CAT (0)-spacesn (2006)
  • Author(s): Fujiwara, Koji, Shioya, Takashi, Koichi Nagano
  • Journal Title: Comment. Math. Helv. 81 Pages: 1305-335
• [Journal Article] Variational convergence of finite networks (2006)
  • Author(s): Kasue, Atsushi
  • Journal Title: Interdiscip. Inform. Sci. 12 Pages: 57-70
  • NAID: 110004321831
• [Journal Article] Convergence of Riemannian manifolds and Laplace operators. II (2006)
  • Author(s): Kasue, Atsushi
  • Journal Title: Potential Anal. 24 Pages: 137-194
• [Journal Article] Metric Riemannian geometry. (2006)
  • Author(s): Fukaya, Kenji
  • Journal Title: Handbook of differential geometry. Vol. II Pages: 189-313
• [Journal Article] Metric Riemannian geometry (2006)
  • Author(s): K.Fukaya
  • Journal Title: Handbook of differential geometry II II Pages: 189-313
• [Journal Article] Volume collapsed three-manifolds with a lower curvature bound (2005)
  • Author(s): T.Shioya and T.Yamaguchi
  • Journal Title: Math. Annalen 333 Pages: 131-155
  • Peer Reviewed
• [Journal Article] Multivalued Morse theory, asymptotic analysis and mirror symmetry. Graphs and patterns in mathematics and theoretical physics (2005)
  • Author(s): Fukaya, Kenji
  • Journal Title: Proc. Sympos. Pure Math(Amer. Math. Soc., Providence, RI) 73 Pages: 205-278
  • Peer Reviewed
• [Journal Article] Miyaoka-Yau inequality and complex hyperbolicity. Topics in almost Hermitian geometry and related fields (2005)
  • Author(s): Itoh, Mitsuhiro
  • Journal Title: World Sci. Publ., Hackensack, NJ Pages: 105-112
  • Peer Reviewed
• [Journal Article] On a conjecture of Wood (2005)
  • Author(s): Kawamura, Kazuhiro
  • Journal Title: Glasg. Math. J. 47no. 1 Pages: 1-5
  • Peer Reviewed
• [Journal Article] Valov, V. On $C\sp ^*$-algebras with the approximate $n$-th root property (2005)
  • Author(s): Chigogidze, A. ; Karasev, A. ; Kawamura, K.
  • Journal Title: Bull. Austral. Math. Soc 72(no. 2) Pages: 197-212
  • Peer Reviewed
• [Journal Article] Tymchatyn, E. D. Hereditarily indecomposable inverse limits of graphs (2005)
  • Author(s): Kawamura, K. ; Tuncali, H. M.
  • Journal Title: Fund. Math 185no. 3 Pages: 195-210
  • Peer Reviewed
• [Journal Article] 4-dimensional almost Kahler manifolds and $L\sp 2$-scalar curvature functional (2005)
  • Author(s): Satoh, Hiroyasu
  • Journal Title: Differential Geom. Appl 23no.2 Pages: 114-127
  • Peer Reviewed
• [Journal Article] Volume collapsed three-manifolds with a lower curv. (2005)
  • Author(s): T.Yamaguchi
  • Journal Title: Math.Annalen 333 Pages: 131-155
• [Journal Article] Miyaoka-Yau inequality and complex geometry (2005)
  • Author(s): M.Itoh
  • Journal Title: Topics in almost Hermition geometry Pages: 105-112
• [Journal Article] The modified Yamabe problem and geometry (2005)
  • Author(s): M.Itoh
  • Journal Title: J.Geom.Anal. 15 Pages: 63-81
• [Journal Article] Hereditarily indecomposable inverse limits of graphs (2005)
  • Author(s): K.Kawamura
  • Journal Title: Fund.Math. 185・3 Pages: 195-210
• [Journal Article] 4-dimensional almost Kahler manifolds (2005)
  • Author(s): H.Satoh
  • Journal Title: Differential Geom.Appl. 23・2 Pages: 114-127
• [Journal Article] Variational convergence over metric spaces
  • Author(s): Kuwae, Kazuhiro, Shioya
  • Journal Title: Trans. of AMS (印刷中)(to appear in)
  • NAID: 120003780146
• [Presentation] Collapsing and essential covering of Riemannian manifolds, (2008)
  • Author(s): T. Yamaguchi
  • Organizer: 9th Pacific Rim Geometry Conference
  • Place of Presentation: Taipei
  • Year and Date: 2008-12-12
• [Presentation] Collapsing and essential covering of Riemannian manifolds, Global KMS International Conference (2008)
  • Author(s): T. Yamaguchi
  • Organizer: Invited address
  • Place of Presentation: Jeju,Korea
  • Year and Date: 2008-10-24
• [Presentation] Convergence of surfaces with bounded total absolute curvature, Analysis seminar (2008)
  • Author(s): T. Yamaguchi
  • Organizer: Kings College,
  • Place of Presentation: London, England
  • Year and Date: 2008-10-16
• [Presentation] Collapsing and essential covering of Riemannian manifolds, 9th Pacific Rim Geometry Conference (2008)
  • Author(s): T. Yamaguchi
  • Organizer: Invited address,National Taiwan University
  • Place of Presentation: Taipei,Taiwan, Dec
  • Year and Date: 2008-10-14
• [Presentation] Perelman による幾何化予想解決 (2008)
  • Author(s): 山口孝男
  • Organizer: 日本数学会・日本物理学会合同講演会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-22
• [Presentation] Geometric analysis on Alexandrov spaces (2008)
  • Author(s): T. Shioya
  • Organizer: spaces,/Seminaire Borel : New approaches to curvature
  • Place of Presentation: スイス レディアブラレ, 4回連続講演
• [Presentation] Functions of finite Dirichlet sum and compactifications of infinite graphs,The 1st MSJ-SI (2008)
  • Author(s): A.Kasue
  • Organizer: Probabilistic Approach to Geometry
  • Place of Presentation: Kyoto
• [Presentation] SLaplacian on Alexandrov spaces (2007)
  • Author(s): T.Shioya
  • Organizer: Stochastic calculus on manifolds, graphs, and random structures
  • Place of Presentation: ドイツハウスドルフ数学研究所
• [Presentation] Two-dimensional Alexandrov spaces with curvature bounded above,International Conference (2007)
  • Author(s): T. Yamaguchi
  • Organizer: Russian-German Geometry Meeting dedicated to the 95th-anniversary of A.D.Alexandrov
  • Place of Presentation: St. Petersburg, Russia
• [Presentation] Two-dimensional Alexandrov spaces with curvature bounded above (2007)
  • Author(s): Yamaguchi, Takao
  • Organizer: Russian-German Geometry Meeting dedicated to the 95th-anniversary of A. D. Alexandrov
  • Place of Presentation: St. Petersburg, Russia.
• [Presentation] CAT(1)空間に対する球面定理 (2006)
  • Author(s): 永野幸一
  • Organizer: 日本数学会2006年度秋季総合分科会, 幾何学分科会特別講演
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2006-09-21
• [Presentation] CAT(1)空間の位相正則性 (2006)
  • Author(s): 永野幸一
  • Organizer: 第53回幾何学シンポジウム, 全体講演
  • Place of Presentation: 金沢大学
  • Year and Date: 2006-08-05
• [Presentation] Laplacian on Alexandrov spaces (2006)
  • Author(s): T.Shioya
  • Organizer: 日中友好幾何学研究集会
  • Place of Presentation: 中国昆明
• [Presentation] An upper curvature bound and singularities (2006)
  • Author(s): T. Yamaguchi
  • Organizer: Singularities, Hayashibara forum, IHES, Bure-sur-Yvett
  • Place of Presentation: France
• [Presentation] 3次元多様体の崩壊 (2006)
  • Author(s): 塩谷隆
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 大阪市立大学,幾何学賞授賞講演
• [Presentation] Collapsing $4$-manifolds under a lower curvature bound, I-IX (series of lectures) (2006)
  • Author(s): T. Yamaguchi
  • Organizer: University of Maryland,College Park
  • Place of Presentation: USA
• [Book] 3次元トポロジーの新展開---リッチフローとポアンカレ予想 (2007)
  • Author(s): 戸田正人
  • Total Pages: 200
  • Publisher: サイエンス社