Classification problems of biharmonic submanifolds

• Project/Area Number: 25887044
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Shimane University (2014); Shumei University (2013)
• Principal Investigator: MAETA Shun 島根大学, 総合理工学研究科(研究院), 講師 (00709644)
• Project Period (FY): 2013-08-30 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 2重調和部分多様体 / 2重調和写像 / 調和写像 / 極小部分多様体 / k重調和写像 / Chen予想 / 完備リーマン多様体 / 一般化されたChen予想

Outline of Final Research Achievements

We studied polyharmonic maps of order k from a complete Riemannian manifold into a non-positively curved manifold. We got the following results.

We showed that biharmonic maps are harmonic under an integral condition. We applied this method into polyharmonic maps of order k. As a joint work with Prof. Urakawa and Prof. Nakauchi, we showed that triharmonic isometric immersion which have finite the 4-energy and finite the L4-norm of the tension field are harmonic. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.

Research Products

• [Journal Article] Triharmonic isometric immersions into a manifold of non-positively constant curvature (2015)
  • Author(s): Shun Maeta, Nobumitsu Nakauchi and Hajime Urakawa
  • Journal Title: Monatshefte fur Mathematik Volume: 未定 Issue: 4 Pages: 551-567
  • DOI: 10.1007/s00605-014-0713-4
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Polyharmonic maps of order k with finite L^p k-energy into Euclidean spaces (2015)
  • Author(s): Shun Maeta
  • Journal Title: Proceedings of the American Mathematical Society Volume: 143 Issue: 5 Pages: 2227-2234
  • DOI: 10.1090/s0002-9939-2014-12382-3
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold (2014)
  • Author(s): Shun Maeta
  • Journal Title: Annals of Global Analysis and Geometry Volume: 未定 Issue: 1 Pages: 75-85
  • DOI: 10.1007/s10455-014-9410-8
  • Peer Reviewed
• [Journal Article] Polyharmonic maps of order k with finite $L^p$ k-energy into Euclidean spaces, (2014)
  • Author(s): Shun Maeta
  • Journal Title: Proceedings of the American Mathematical Society Volume: 未定
  • Peer Reviewed
• [Presentation] Biharmonic submanifolds and triharmonic submanifolds (2014)
  • Author(s): Shun Maeta
  • Organizer: International Workshop on Mathematical Sciences in Matsue
  • Place of Presentation: Shimane University
  • Year and Date: 2014-10-12
  • Invited
• [Presentation] Biharmonic submanifolds, triharmonic submanifolds and Chen's conjecture (2014)
  • Author(s): Shun Maeta
  • Organizer: Interenational Workshop on Finite Type Submanifolds
  • Place of Presentation: Istanbul Technical University
  • Year and Date: 2014-09-04
  • Invited
• [Presentation] 2重調和部分多様体とChen予想 (2014)
  • Author(s): 前田瞬
  • Organizer: 松江セミナー
  • Place of Presentation: 島根大学
  • Year and Date: 2014-07-02
  • Invited
• [Presentation] Chen予想と3重調和部分多様体 (2014)
  • Author(s): 浦川 肇，中内 伸光，前田 瞬
  • Organizer: 日本数学会2014年度年会
  • Place of Presentation: 学習院大学
• [Presentation] Biharmonic submanifolds and Chen's conjecture (2013)
  • Author(s): Shun Maeta
  • Organizer: Conferinta of Universitatea Alexandru loan Cuza
  • Place of Presentation: Universitatea Alexandru loan Cuza