Study on the construction of hypersurfaces with constant scalar curvature by using geometric analysis

• Project/Area Number: 19540062
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Ibaraki University
• Principal Investigator: OKAYASU Takashi 茨城大学, 教育学部, 教授 (00191958)
• Co-Investigator(Kenkyū-buntansha): YAGITA Nobuaki 茨城大学, 教育学部, 教授 (20130768)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: スカラー曲率 / 超曲面 / 極小部分多様体 / ユークリッド空間 / 比較定理 / 第二基本形式

Research Abstract

The classification problem of complete surfaces with constant Gaussian curvature in the 3-dimensional Euclidean space was solved completely in old days. If we generalize this problem to higher dimensions, the problem is to classify complete hypersurfaces with constant scalar curvature. This probelem is very difficult because of lack of examples. Our study shows that there are many new examples of complete hypersurfaces with constant positive scalar curvature.

Research Products

• [Journal Article] Closed hypersurfaces in Riemannian manifolds whose second fundamental forms are of constant length (2011)
  • Author(s): 岡安隆
  • Journal Title: AIP conference proceedings : International workshop on complex structures, integrability and vector fields Volume: Vol.1340 Pages: 90-96
  • Peer Reviewed
• [Journal Article] Closed hypersurfaces in Riemannian manifolds whose second fundamental forms are of constant length (2011)
  • Author(s): Takashi Okayasu
  • Journal Title: International workshop on complex structures, integrability and vector fields Volume: 1340 Pages: 90-96
  • Peer Reviewed
• [Journal Article] New examples of complete hypersurfaces with constant positive scalar curvature in the Euclidean space (2010)
  • Author(s): 岡安隆
  • Journal Title: J. Math. Soc. Japan Volume: Vol.62, No.4 Pages: 1137-1166
  • NAID: 10027871057
  • Peer Reviewed
• [Presentation] ユークリッド空間の完備極小部分多様体の構成についての2つの結果 (2011)
  • Author(s): 岡安隆
  • Organizer: RIMS研究集会部分多様体の微分幾何学的研究
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-06-29
• [Presentation] ユークリッド空間の完備極小部分多様体の構成についての2つの結果 (2011)
  • Author(s): 岡安隆
  • Organizer: RIMS研究集会部分多様体の微分幾何学的研究
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2011-06-29
• [Presentation] Comparison theorems for ODEs and their application to geometry of Weingarten hypersurfaces (2011)
  • Author(s): 岡安隆
  • Organizer: The International Conference of Differential Geometry and Tanaka Theory
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-01-25
• [Presentation] Comparison theorems for ODEs and their application to geometry of Weingarten hypersurfaces (2011)
  • Author(s): 岡安隆
  • Organizer: The International Conference of Differential Geometry and Tanaka Theory
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2011-01-25
• [Presentation] A comparison theorem for an ODE and its application to geometry of Weingarten hypersurfaces (2010)
  • Author(s): 岡安隆
  • Organizer: 10th International Workshop Complex structures, integrability and vector fields
  • Place of Presentation: ブルガリア科学アカデミー(ブルガリア)
  • Year and Date: 2010-09-15
• [Presentation] Riemann多様体での第二基本形式の長さが一定な超曲面の存在について (2009)
  • Author(s): 岡安隆
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] Riemann 多様体での第二基本形式の長さが一定な超曲面の存在について (2009)
  • Author(s): 岡安隆
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] Riemann多様体における第二基本形式の長さが一定な閉超曲面の存在について (2009)
  • Author(s): 岡安隆
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] A construction of complete hypersurfaces with constant scalar curvature in the Euclidean space (2008)
  • Author(s): 岡安隆
  • Organizer: The ninth Pacific Rim Geometry Conference(第9回環太平洋幾何学研究集会)
  • Place of Presentation: 国立台湾大学、台湾
  • Year and Date: 2008-12-10
• [Presentation] A construction of complete hypersurfaces with constant scalar curvature in the Euclidean space (2008)
  • Author(s): 岡安隆
  • Organizer: The Ninth Pacific Rim Geometry Conference
  • Place of Presentation: National Taiwan University, Taipei, Taiwan
  • Year and Date: 2008-12-10
• [Presentation] A construction of complete hypersurfaces with constant scalar curvature in the Euclidean space (2008)
  • Author(s): 岡安隆
  • Organizer: The ninth Pacific Rim Geometry Conference (第9回環太平洋幾何学研究集会)
  • Place of Presentation: 国立台湾大学
  • Year and Date: 2008-12-01
• [Presentation] Riemann多様体における第二基本形式の長さ一定な超曲面の存在について (2008)
  • Author(s): 岡安隆
  • Organizer: 新潟微分幾何学研究集会
  • Place of Presentation: 新潟大学
  • Year and Date: 2008-11-06
• [Presentation] Riemann 多様体における第二基本形式の長さ一定な超曲面の存在について (2008)
  • Author(s): 岡安隆
  • Organizer: 新潟微分幾何学研究集会
  • Place of Presentation: 新潟大学
  • Year and Date: 2008-11-06
• [Presentation] A construction of complete hypersurfaces with constant scalar curvature (2007)
  • Author(s): 岡安隆
  • Organizer: 10th International Conference on Differential Geometry and Its Applications
  • Place of Presentation: Palacky University,チェコ
  • Year and Date: 2007-08-31
• [Presentation] A construction of complete hypersurfaces with constant scalar curvature (2007)
  • Author(s): 岡安隆
  • Organizer: 10th International Conference on Differential Geometry and Its Applications
  • Place of Presentation: Palacky University
  • Year and Date: 2007-08-31
• [Presentation] A construction of complete hypersurfaces with constant scalar curvature (2007)
  • Author(s): Takashi Okayasu
  • Organizer: 10th International Conference on Differential Geometry and Its Applications
  • Place of Presentation: Palacky University, Olomouc, Czech Republic
  • Year and Date: 2007-08-31
• [Remarks]
  • URL: http://info.ibaraki.ac.jp/scripts/websearch/index.htm
• [Remarks]
  • URL: http://info.ibaraki.ac.jp/scripts/websearch/index.htm
• [Remarks] 茨城大学研究者情報総覧
  • URL: http://info.ibaraki.ac.jp/scripts/websearch/index.htm