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Challenging study on self-shrinkers of mean curvature flow and applications

Research Project

Project/Area Number 25610016
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionFukuoka University

Principal Investigator

Cheng Qing-Ming  福岡大学, 理学部, 教授 (50274577)

Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords平均曲率フロー / 最大値原理 / 部分多様体 / セルフ-シュリンカー / 面積汎関数の変分 / せルフ-シュリンカー / 部分多様体の幾何学 / 固有値理論 / 安定性作用素
Outline of Final Research Achievements

We generalize the maximum principle due to Omori and Yau for Laplacian to L operator on complete self-shrinkers of mean curvature flow. Making use of this generalized maximum principle for L operator, we study complete self-shrinkers of mean curvature and give a complete classification for 2-dimensional complete self-shrinkers in the 3-dimensional Euclidean space. Furthermore, for complete self-shrinkers of mean curvature flow with polynomial area growth, we get the second pinching constant of the constant length of the second fundamental form. We also prove a universal inequality on eigenvalues of L operator in complete self-shrinkers. According to this universal inequality, we obtain an upper bound and a lower bound of eigenvalues for L operator in complete self-shrinkers.

Report

(4 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (30 results)

All 2016 2015 2014 2013 Other

All Int'l Joint Research (1 results) Journal Article (8 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 8 results,  Open Access: 6 results,  Acknowledgement Compliant: 6 results) Presentation (17 results) (of which Int'l Joint Research: 4 results,  Invited: 17 results) Book (1 results) Remarks (2 results) Funded Workshop (1 results)

  • [Int'l Joint Research] 華南師範大学(中国)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Rigidity theorems of λ-hypersurfaces2016

    • Author(s)
      Qing-Ming Cheng, S. Ogata and G. Wei
    • Journal Title

      Comm. Analy. Geom.

      Volume: in press

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] 2-dimensional complete self-shrinkers in R32016

    • Author(s)
      Qing-Ming Cheng and S. Ogata
    • Journal Title

      Math. Z.

      Volume: in press Issue: 1-2 Pages: 537-542

    • DOI

      10.1007/s00209-016-1665-2

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Complete self-shrinkers of the mean curvature flow2015

    • Author(s)
      Qing-Ming Cheng and Y. Peng
    • Journal Title

      Calculus of Variations and PDEs.

      Volume: 52 Issue: 3-4 Pages: 497-506

    • DOI

      10.1007/s00526-014-0720-2

    • Related Report
      2015 Annual Research Report 2014 Research-status Report 2013 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A gap theorem of self-shrinkers2015

    • Author(s)
      Qing-Ming Cheng and Wei
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 367 Pages: 4895-4915

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A gap theorem of self-shrinkers2015

    • Author(s)
      Qing-Ming Cheng and Guoxin Wei
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: in press

    • Related Report
      2014 Research-status Report 2013 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Critical points of the weighted area functional2015

    • Author(s)
      Qing-Ming Cheng
    • Journal Title

      Springer Proceedings in Mathematics and Statistics, Geometry and Topology of Manifolds

      Volume: Proceedings

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Estimates for eigenvalues of L operator on self-shrinkers2013

    • Author(s)
      Qing-Ming Cheng and Y. Peng
    • Journal Title

      Comm. Contemporary Math.

      Volume: 15 Issue: 06 Pages: 1350011-1350011

    • DOI

      10.1142/s0219199713500119

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Self-shrinkers of the mean curvature flow2013

    • Author(s)
      Qing-Ming Cheng and Y. Peng
    • Journal Title

      Differential geometry of submanifolds and its related topics, World Scientific

      Volume: Proceddings Pages: 147-163

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] Complete self-shrinkers2016

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      Seminar on Mathematics, Guangxi Normal University
    • Place of Presentation
      Guangxi Normal University, China
    • Year and Date
      2016-03-25
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of λ-hypersurfaces of the weighted volume-preserving mean curvature flow2016

    • Author(s)
      成 慶明
    • Organizer
      日本数学会年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-16
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Geometry of critical points of weighted area functional,2015

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      School & Workshop on Geometric Analysis, KIAS
    • Place of Presentation
      KIAS, Korea
    • Year and Date
      2015-12-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 汎関数の臨界点の幾何学2015

    • Author(s)
      成 慶明
    • Organizer
      広島幾何学研究集会 2015
    • Place of Presentation
      広島大学
    • Year and Date
      2015-10-07
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Geometry of λ-hypersurfaces2015

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      International conference on geometry and topology of submanifolds
    • Place of Presentation
      Henan Normal University, China
    • Year and Date
      2015-09-25
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of critical points of functional2015

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      Seminar on Mathematics, Wuhan University
    • Place of Presentation
      Wuhan University, China
    • Year and Date
      2015-06-23
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 重み付き体積を保つ平均曲率フロー2015

    • Author(s)
      成 慶明
    • Organizer
      研究集会「リーマン幾何と幾何解析」
    • Place of Presentation
      筑波大学
    • Year and Date
      2015-03-06 – 2015-03-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Critical points of the weighted area functional2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      The 10th Geometry Conference for the Friendship of China and Japan
    • Place of Presentation
      Fudan University, Shanghai, China
    • Year and Date
      2014-09-06 – 2014-09-12
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] From self-shrinkers to λ-hypersurfaces2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      Beijing Normal University
    • Place of Presentation
      Beijing Normal University, Beijing, China
    • Year and Date
      2014-08-17
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the weighted area functional2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      Nankai University
    • Place of Presentation
      Nankai University, Tianjin, China
    • Year and Date
      2014-08-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] The rigidity of complete self-shrinkers of the mean curvature flow2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      South China Normal University
    • Place of Presentation
      South China Normal University, Guangzhou, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Self-shrinkers of mean curvature flow2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      International Workshop on Differential Geometry 2014
    • Place of Presentation
      Hotel New Imari, Imari
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] A survey on complete self-shrinkers2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      Beijing Normal University
    • Place of Presentation
      Beijing Normal University, Beijing, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Critical points of the weighted volume functional2014

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      Fujian Normal Universty
    • Place of Presentation
      Fujian Normal Universty, Fuzhou, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Universal estimates for eigenvalues and applications2013

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      The Sixth International Congress of Chinese Mathematics
    • Place of Presentation
      National Taiwan University, Taipei, Taiwan
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 平均曲率フローの完備セルフーシュリンカーについて2013

    • Author(s)
      成 慶明
    • Organizer
      九州大学幾何学セミナー
    • Place of Presentation
      九州大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Complete self-shrinkers of the mean curvature flow2013

    • Author(s)
      Qing-Ming Cheng
    • Organizer
      The Eleventh Pacific Rim Geometry Conference
    • Place of Presentation
      Fudan University, Shanghai, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Book] Differential geometry of submanifolds and its related topics2013

    • Author(s)
      S. Maeda, Y. Ohnita and Q.-M. Cheng
    • Total Pages
      289
    • Publisher
      World Scientific
    • Related Report
      2013 Research-status Report
  • [Remarks]

    • URL

      http://www.cis.fukuoka-u.ac.jp/~cheng/

    • Related Report
      2015 Annual Research Report
  • [Remarks] Home Page of Cheng Qing-Ming

    • URL

      http://www.cis.fukuoka-u.ac.jp/~cheng/

    • Related Report
      2014 Research-status Report
  • [Funded Workshop] Japan-China Geometry Conference2015

    • Place of Presentation
      京都大学と奈良女子大学
    • Year and Date
      2015-09-06
    • Related Report
      2015 Annual Research Report

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Published: 2014-07-25   Modified: 2019-07-29  

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