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Geometric and Global Analysis of Scalar Curvature and Einstein Metrics

Research Project

Project/Area Number 24340008
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Geometry
Research InstitutionTokyo Institute of Technology (2013-2017)
Tohoku University (2012)

Principal Investigator

AKUTAGAWA KAZUO  東京工業大学, 理学院, 教授 (80192920)

Co-Investigator(Renkei-kenkyūsha) FUTAKI Akito  東京大学, 数理科学研究科, 教授 (90143247)
KOBAYASHI Osamu  大阪市立大学, 数学研究所, 特別研究員 (10153595)
FURUTA Mikio  東京大学, 数理科学研究科, 教授 (50181459)
NAYATANI Shin  名古屋大学, 多元数理科学研究科, 教授 (70222180)
ONO Hajime  埼玉大学, 理工学部, 准教授 (70467033)
HONDA Shouhei  東北大学, 理学研究科, 准教授 (60574738)
MATSUO Shinichiroh  名古屋大学, 多元数理科学研究科, 准教授 (40599487)
MATSUMOTO Yoshihiko  大阪大学, 理学研究科, 助教 (00710625)
Research Collaborator Carron Gilles  Université de Nantes, Département de Mathématiques, Professor
Mazzeo Rafe  Stanford University, Department of Mathematics, Professor
Mondello Ilaria  Université de Nantes, Département de Mathématiques, Researcher
Vertman Boris  Universität Münster, Mathematisches Institut, Professor
Project Period (FY) 2012-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥17,810,000 (Direct Cost: ¥13,700,000、Indirect Cost: ¥4,110,000)
Fiscal Year 2016: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2015: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2014: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2013: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2012: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Keywords微分幾何 / 幾何解析 / スカラー曲率 / アインシュタイン計量 / 山辺不変量 / 国際研究交流 / 国際情報交換 / 多国籍
Outline of Final Research Achievements

On a compact manifold with very general singularities, we have studied the Yamabe problem and have established a generalization of Aubin’s inequality for Yamabe constants. When the inequality is strict, we have proved the existence of singular Yamabe metrics.
When the equality of the inequality holds, we have constructed an example of singular manifolds which have not singular Yamabe metrics. For an edge-cone Einstein metric on a smooth manifold, we have constructed an appropriate family of smooth metrics with Ricci curvature bounded below by the Einstein constant. As a corollary, we have obtained an estimate of the Yamabe invariant from below by using the existence of edge-cone Einstein metrics.

Report

(7 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • Research Products

    (49 results)

All 2020 2018 2017 2016 2015 2014 2013 2012 Other

All Int'l Joint Research (4 results) Journal Article (13 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 11 results,  Open Access: 2 results,  Acknowledgement Compliant: 4 results) Presentation (28 results) (of which Int'l Joint Research: 4 results,  Invited: 24 results) Book (2 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Stanford University(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Indian Institute of Science, Bangalore(インド)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] スタンフォード大学(米国)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] ナント大学(フランス)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] The Yamabe invariant2020

    • Author(s)
      K. Akutagawa
    • Journal Title

      Amer. Math. Soc. Transl., Sugaku Expositions

      Volume: -

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Yamabe problem on Dirichlet spaces2017

    • Author(s)
      K. Akutagawa, G. Carron and R. Mazzeo
    • Journal Title

      Tsinghua Lectures in Mathematics

      Volume: -

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Edge-cone Einstein metrics and the Yamabe invariant2017

    • Author(s)
      Kazuo Akutagawa
    • Journal Title

      Mathematisches Forschungsinstitut Oberwolfach, Report No. 36

      Volume: 36 Issue: 3 Pages: 43-46

    • DOI

      10.4171/owr/2017/36

    • Related Report
      2017 Annual Research Report
  • [Journal Article] Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space2017

    • Author(s)
      Kazuo Akutagawa, Reiko Aiyama, Satoru Imagawa, Yu Kawakami
    • Journal Title

      Annali di matematica Pure ed Applicata

      Volume: -

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Minimal Legendrian surfaces in the five-dimensional Heisenberg group2016

    • Author(s)
      Kazuo Akutagawa, Reiko Aiyama
    • Journal Title

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

      Volume: 154 Pages: 1-13

    • DOI

      10.1007/978-4-431-56021-0_1

    • ISBN
      9784431560197, 9784431560210
    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Edge-cone Einstein計量と山辺不変量2016

    • Author(s)
      芥川和雄
    • Journal Title

      第63回幾何学シンポジウム予稿集(基調講演)

      Volume: 63 Pages: 101-113

    • Related Report
      2016 Annual Research Report
    • Acknowledgement Compliant
  • [Journal Article] Proper harmonic maps between asymptotically hyperbolic manifolds2016

    • Author(s)
      Kazuo Akutagawa, Yoshihiko Matsumoto
    • Journal Title

      Mathematische Annallen

      Volume: 364 Issue: 3-4 Pages: 793-811

    • DOI

      10.1007/s00208-015-1229-5

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Holder regularity of solutions for Schrodinger operators on stratified spaces2015

    • Author(s)
      Kazuo Akutagawa, Gilles Carron, Rafe Mazzeo
    • Journal Title

      Journal of Functional Analysis

      Volume: - Issue: 3 Pages: 815-840

    • DOI

      10.1016/j.jfa.2015.02.003

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] 山辺不変量2014

    • Author(s)
      芥川 和雄
    • Journal Title

      数学・論説(日本数学会)

      Volume: 66 Pages: 31-60

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 山辺不変量2013

    • Author(s)
      芥川 和雄
    • Journal Title

      数学・論説

      Volume: (掲載確定)(印刷中)

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Aubin's lemma for the Yamabe constants of infinite coverings and a positive mass theorem2012

    • Author(s)
      Kazuo Akutagawa
    • Journal Title

      Mthematische Annalen

      Volume: 352 Issue: 4 Pages: 829-864

    • DOI

      10.1007/s00208-011-0667-y

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Computations of the orbifold Yamabe invariant2012

    • Author(s)
      Kazuo Akutagawa
    • Journal Title

      Math. Zeit.

      Volume: 271 Pages: 611-625

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 3-manifolds with positive flat conformal structure2012

    • Author(s)
      Reiko Aiyama, Kazuo Akutagawa
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: (未定)(掲載決定) Issue: 10 Pages: 3587-3592

    • DOI

      10.1090/s0002-9939-2012-11423-6

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Presentation] A gap theorem for positive Einstein metrics on the four-sphere2018

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Stanford大学「Geometry Seminar」
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A gap theorem for positive Einstein metrics on the four-sphere2017

    • Author(s)
      芥川 和雄
    • Organizer
      福岡大学微分幾何セミナー
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Gap theorems for positive Einstein metrics on four-manifolds2017

    • Author(s)
      芥川和雄
    • Organizer
      東工大幾何セミナー
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invarinat2017

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Analysis, Geometry and Topology of Positive Scalar Curvature Metrics
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A gap theorem for positive Einstein metrics on the four-sphere2017

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Special Session :「Differential Geometry」of theThird Congress of the Pacific Rim Mathematical Association Oaxaca
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invarinat2017

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      日本数学会―台湾数学会 ジョイントミーティング,Geometry Session
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A gap theorems for positive Einstein metrics on the 4-sphere2017

    • Author(s)
      芥川和雄
    • Organizer
      大阪大学理学部「幾何セミナー」
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invarinat2017

    • Author(s)
      芥川和雄
    • Organizer
      大阪大学理学部「談話会」
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invariant2016

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      第2回日中幾何学研究集会
    • Place of Presentation
      福建師範大学(福州市・中国)
    • Year and Date
      2016-09-11
    • Related Report
      2016 Annual Research Report
  • [Presentation] Edge-cone Einstein計量と山辺不変量2016

    • Author(s)
      芥川和雄
    • Organizer
      第63回幾何学シンポジウム、日本数学会・幾何学分科会
    • Place of Presentation
      岡山大学(岡山市北区)
    • Year and Date
      2016-08-29
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invariant2016

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Seminar on Geometry and Topology
    • Place of Presentation
      University of Luxembourg (Luxembourg)
    • Year and Date
      2016-06-20
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] 山辺の問題とその発展:特異空間上の山辺不変量とリッチフローとの関係2016

    • Author(s)
      芥川和雄
    • Organizer
      基礎自然科学系・学術セミナー
    • Place of Presentation
      岩手大学(岩手県盛岡市)
    • Year and Date
      2016-03-24
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invariant2016

    • Author(s)
      芥川和雄
    • Organizer
      内藤博夫先生退官記念研究集会
    • Place of Presentation
      山口大学(山口県山口市)
    • Year and Date
      2016-03-06
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invariant2016

    • Author(s)
      芥川和雄
    • Organizer
      東工大幾何セミナー
    • Place of Presentation
      東京工業大学(東京都目黒区)
    • Year and Date
      2016-01-08
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Edge-cone Einstein metrics and the Yamabe invariant2015

    • Author(s)
      芥川和雄
    • Organizer
      数学教室談話会
    • Place of Presentation
      京都大学・大学院理学研究科(京都市左京区)
    • Year and Date
      2015-12-02
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] 山辺不変量とsingular Einstein計量---小林プログラムについて---2015

    • Author(s)
      芥川和雄
    • Organizer
      秋葉原微分幾何セミナー
    • Place of Presentation
      首都大学東京・秋葉原サテライトキャンパスAB(東京都千代田区)
    • Year and Date
      2015-07-04
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] The Yamabe invariant and singular Einstein metrics2015

    • Author(s)
      芥川和雄
    • Organizer
      研究集会「リーマン幾何と幾何解析」
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2015-03-06
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The Yamabe invariant and singular Einstein metrics2015

    • Author(s)
      芥川和雄
    • Organizer
      東北大学・大学院情報科学研究科「幾何解析セミナー」
    • Place of Presentation
      東北大学(宮城県仙台市青葉区)
    • Year and Date
      2015-01-26
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The Yamabe problem on Dirichlet spaces2014

    • Author(s)
      芥川和雄
    • Organizer
      研究集会「確率論と幾何学」
    • Place of Presentation
      東京工業大学(東京都目黒区)
    • Year and Date
      2014-09-24
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The Yamabe problem on stratified spaces2013

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Geometric Analysis Workshop, Tsinghua-Sanya Mathematical Forum
    • Place of Presentation
      中国,三亜(招待講演)
    • Year and Date
      2013-01-06
    • Related Report
      2012 Annual Research Report
  • [Presentation] The Yamabe problem on manifolds with conic singularities2012

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Geometric PDEs
    • Place of Presentation
      フランス,ポアンカレ研究所(招待講演)
    • Year and Date
      2012-11-06
    • Related Report
      2012 Annual Research Report
  • [Presentation] The Yamabe problem on manifolds with conic singularities2012

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      第8回日中友好幾何学研究集会
    • Place of Presentation
      中国,四川大学(招待講演)
    • Year and Date
      2012-09-08
    • Related Report
      2012 Annual Research Report
  • [Presentation] The Yamabe problem on singular spaces

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      The Asian Mathematical Conference 2013
    • Place of Presentation
      Busan Exhibition and Convention Center (BEXCO), Busan, Korea
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 特異空間およびDirichlet空間上の山辺の問題

    • Author(s)
      芥川 和雄
    • Organizer
      東京工業大学・数学教室・大岡山談話会
    • Place of Presentation
      東京工業大学(東京都)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The Yamabe problem on singular spaces and Dirichlet spaces

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      The 8th Pacific Rim Complex Geometry Conference
    • Place of Presentation
      Dalian University of Technology, Dalian City, China
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 特異空間上の山辺の問題,Aubinの不等式,edge-cone Einstein計量

    • Author(s)
      芥川 和雄
    • Organizer
      福岡大学微分幾何セミナー
    • Place of Presentation
      福岡大学(福岡県)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The Yamabe problem on edge-cone manifolds and Aubin’s inequality

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Geometric Invariance and Nonlinear Partial Differential Equations
    • Place of Presentation
      Kioloa Coastal Campus, Australian National University, Kioloa, Australia
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The Yamabe problem on edge-cone manifolds and Aubin’s inequality

    • Author(s)
      Kazuo Akutagawa
    • Organizer
      Geometry Seminar
    • Place of Presentation
      Stanford University, Stanford, USA
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Book] 山辺の問題と山辺不変量(幾何解析,幾何学百科第4巻)2018

    • Author(s)
      芥川和雄, 小林治
    • Total Pages
      85
    • Publisher
      朝倉書店
    • Related Report
      2017 Annual Research Report
  • [Book] 山辺の問題2013

    • Author(s)
      小林治,芥川和雄,井関裕靖
    • Total Pages
      75
    • Publisher
      日本数学会
    • Related Report
      2013 Annual Research Report
  • [Funded Workshop] Geometric Analysis in Geometry and Topology 20162016

    • Place of Presentation
      東京理科大学(東京都新宿区)および東京工業大学(東京都目黒区)
    • Year and Date
      2016-12-12
    • Related Report
      2016 Annual Research Report
  • [Funded Workshop] Geometric Analysis in Geometry and Topology 20152015

    • Place of Presentation
      東京理科大学森戸記念館(東京都新宿区)
    • Year and Date
      2015-10-09
    • Related Report
      2015 Annual Research Report

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Published: 2012-04-24   Modified: 2023-03-16  

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