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Geometric analysis of Lagrangian mean curvature flows and Ricci flows

Research Project

Project/Area Number 16H07229
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionTokyo University of Science

Principal Investigator

Yamamoto Hikaru  東京理科大学, 理学部第一部数学科, 助教 (50778173)

Project Period (FY) 2016-08-26 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywordsmean curvature flow / Lagrangian / special Lagrangian / Ricci flow / mirror symmetry / dHYM connection / 特殊ラグランジュ部分多様体 / 変形エルミート・ヤン・ミルズ接続 / ラグランジュ平均曲率流 / リッチフロー / 平均曲率流 / ラグランジュ部分多様体
Outline of Final Research Achievements

I proposed a method to construct a special Lagrange submanifold in the lattice quotient of the tangential bundle of a tropical manifold. I also proved that the Fourier-Mukai transform of this special Lagrange submanifold is a deformed Hermitian Yang Mills connection with support on a complex submanifold in the mirror. A paper on these results will be published in Math. Z.
I proved that if the second fundamental form of a self-shrinker satisfying the pinching condition takes zero at some point, then it becomes a plane. As an application, if a codimension 1 mean curvature flow with initial pinching condition in Euclidean space develops finite time singularities, then all general type I singularities are actually special type I singularities. The proof is summarized in the RIMS Kokyuroku.

Report

(3 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Annual Research Report
  • Research Products

    (17 results)

All 2018 2017 2016 Other

All Int'l Joint Research (1 results) Journal Article (4 results) (of which Peer Reviewed: 4 results) Presentation (12 results) (of which Int'l Joint Research: 5 results,  Invited: 10 results)

  • [Int'l Joint Research] 清華大学(中国)

    • Related Report
      2017 Annual Research Report
  • [Journal Article] Examples of Ricci-Mean Curvature Flows2018

    • Author(s)
      Hikaru Yamamoto
    • Journal Title

      The Journal of Geometric Analysis

      Volume: 28 Issue: 2 Pages: 983-1004

    • DOI

      10.1007/s12220-017-9851-y

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Special Lagrangian and deformed Hermitian Yang-Mills on tropical manifold2018

    • Author(s)
      Hikaru Yamamoto
    • Journal Title

      Mathematische Zeitschrift

      Volume: 印刷中 Issue: 3-4 Pages: 1023-1040

    • DOI

      10.1007/s00209-018-2050-0

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Gauss maps of the Ricci-mean curvature flow2018

    • Author(s)
      Naoyuki Koike, Hikaru Yamamoto
    • Journal Title

      Geometriae Dedicata

      Volume: 194 Issue: 1 Pages: 169-185

    • DOI

      10.1007/s10711-017-0271-8

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lagrangian self-similar solutions in gradient shrinking Kahler-Ricci solitons2017

    • Author(s)
      Hikaru Yamamoto
    • Journal Title

      Journal of Geometry

      Volume: 108(1) Issue: 1 Pages: 247-254

    • DOI

      10.1007/s00022-016-0336-0

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Presentation] On correspondence between special Lagrangian submanifolds and deformed Hermitian Yang-Mills connections2018

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      Geometry of Submanifolds and Integrable Systems
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Mean curvature flows in several ambient spaces and its monotonicity formulas2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      首都大幾何学セミナー
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 平均曲率流の特異点について2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      部分多様体論の潮流
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 特殊ラグランジュ部分多様体と変形エルミート・ヤン・ミルズ接続のミラー対応2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      日本数学会
    • Related Report
      2017 Annual Research Report
  • [Presentation] LYZ ansatz on tropical manifolds2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      The Third Japanese-Spanish workshop on Differential Geometry
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] リッチ平均曲率流とそのガウス写像について2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      福岡大学微分幾何研究会
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Ricci-mean curvature flows and its Gauss maps2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      Geometric flows and Related Problems
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On coupled flows with mean curvature flows and Ricci flows2017

    • Author(s)
      山本 光
    • Organizer
      Workshop on Analysis in Kagurazaka 2017
    • Place of Presentation
      東京理科大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Ricci-mean curvature flows and its Gauss maps2017

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      The 13th OCAMI-RIRCM Joint Differential Geometry Workshop on Submanifold Geometry and Lie Theory
    • Place of Presentation
      大阪市立大学
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ricci-mean curvature flows in gradient shrinking Ricci solitons2016

    • Author(s)
      Hikaru Yamamoto
    • Organizer
      The second China-Japan geometry conference
    • Place of Presentation
      Fujian Normal University
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] リッチフローと平均曲率流の混合方程式について2016

    • Author(s)
      山本 光
    • Organizer
      Okayama Workshop on Partial Differential Equations
    • Place of Presentation
      岡山大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] リッチ平均曲率流とその具体例について2016

    • Author(s)
      山本 光
    • Organizer
      福岡大学微分幾何研究会
    • Place of Presentation
      福岡大学セミナーハウス
    • Related Report
      2016 Annual Research Report
    • Invited

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Published: 2016-09-02   Modified: 2019-03-29  

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