Organization
Faculty of Engineering Science Professor
Title
Professor
Contact information
External link
FUJIOKA,Atsushi
Degree
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博士（数理科学） ( 1996.3 )
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修士（理学） ( 1992.3 )
Research Interests
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微分幾何学
Research Areas
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Natural Science / Geometry
Education
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The University of Tokyo Graduate School, Division of Mathematical Sciences
1992.4 - 1996.3
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The University of Tokyo Graduate School, Division of Natural Science
1990.4 - 1992.3
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The University of Tokyo Faculty of Science
- 1990.3
Research History
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Professor, Kansai University
2012.4
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Associate Professor, Hitotsubashi University
2007.4 - 2012.3
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Assistant Professor, Hitotsubashi University
2003.4 - 2007.3
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Lecturer, Kanazawa University
2000.4 - 2003.3
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Assistant, Kanazawa University
1997.1 - 2000.3
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Research Fellowship for Young Scientists, JSPS
1995.4 - 1996.12
Professional Memberships
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Mathematical Society of Japan
Papers
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Isothermally asymptotic centroaffine minimal surfaces of cohomogeneity one Reviewed
Atsushi Fujioka
International Journal of Geometry 12 56 - 68 2023
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Centroaffine surfaces of cohomogeneity one with planar curvature lines Reviewed
Atsushi Fujioka, Hitoshi Furuhata
Colloquium Mathematicum 172 ( 2 ) 173 - 190 2023
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等積中心アファイン曲線と円周の微分同相群
藤岡敦
数理解析研究所講究録 2210, 85-93 2022
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Centroaffine surfaces of cohomogeneity one Reviewed
Atsushi Fujioka, Hitoshi Furuhata
Bulletin of the Brazilian Mathematical Society, New Series 50, Issue 1, pp. 291-313 2019
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The center map of a centroaffine ruled surface Reviewed
Atsushi Fujioka, Hitoshi Furuhata
Scientific Annals of the Alexandru Ioan Cuza University of Iaşi (New Series). Mathematics Tumul LXIV, f. 2, pp. 343-355 2018
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Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection Reviewed
Atsushi Fujioka, Kunio Hamamoto, Yasunobu Nakai
Contributions to Algebra and Geometry 58, No. 2, pp. 303-310 2017
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Projective surfaces and pre-normalized Blaschke immersions of codimension two Reviewed
Atsushi Fujioka, Hitoshi Furuhata, Takeshi Sasaki
International Electronic Journal of Geomttry 9, No. 1, pp. 100-110 2016
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Projective minimality for centroaffine minimal surfaces Reviewed
Atsushi Fujioka, Hitoshi Furuhata, Takeshi Sasaki
Journal of Geometry 105 ( 1 ) 87 - 102 2014
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Multi-Hamiltonian structures on spaces of closed equicentroaffine plane curves associated to higher KdV flows Reviewed
Alsushi Fujioka, Takashi Kurose
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 10 2014
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Flat centroaffine surfaces with non-semisimple Tchebychev operator Reviewed
Atsushi Fujioka
Differential geometry of submanifolds and its related topics pp. 180-189 180 - 189 2014
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Surfaces in the complexified sphere parametrized by a complex orthogonal net Reviewed
Atsushi Fujioka
JP Journal of Geometry and Topology 10, pp. 89-98 ( 2 ) 89 - 181 2010
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Centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants Reviewed
FUJIOKA,Atsushi
Journal of Mathematical Analysis and Applications 365 ( 2 ) 694 - 700 2010
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Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves Reviewed
Atsushi Fujioka, Takashi Kurose
International Journal of Geometric Methods in Modern Physics 7 ( 1 ) 165 - 175 2010
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Centroaffine minimal surfaces with non-semisimple centroaffine Tchebychev operator Reviewed
Atsushi Fujioka
Results in Mathematics 56 ( 1-4 ) 177 - 195 2009
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Geometry of the space of closed curves in the complex hyperbola Reviewed
Atsushi Fujioka, Takashi Kurose
Kyushu Journal of Mathematics 63 ( 1 ) 161 - 165 2009
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Bonnet surfaces with non-flat normal bundle in the hyperbolic four-space Reviewed
FUJIOKA,Atsushi
Far East Journal of Mathematical Sciences 30 ( 2 ) 381 - 387 2008
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Motions of curves in the complex hyperbola and the Burgers hierarchy Reviewed
Atsushi Fujioka, Takashi Kurose
Osaka Journal of Mathematics 45 ( 4 ) 1057 - 1065 2008
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Timelike surfaces with harmonic inverse mean curvature Reviewed
Atsushi Fujioka, Jun-ichi Inoguchi
Advanced Studies in Pure Mathematics 51 113 - 141 2008
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Deformations of surfaces preserving conformal or similarity invariants Reviewed
Atsushi Fujioka, Jun-ichi Inoguchi
Progress in Mathematics 252 ( 252 ) 53 - 67 2007
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Centroaffine minimal surfaces with constant curvature metric Reviewed
Atsushi Fujioka
Kyungpook Mathematical Journal 46 ( 2 ) 297 - 305 2006
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随伴族をもつ曲面
藤岡敦
一橋論叢 133 ( 3 ) 60 - 78 2005
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Bianchi surfaces with constant chebyshev angle Reviewed
Atsushi Fujioka
Tokyo Journal of Mathematics 27 ( 1 ) 149 - 153 2004
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Bonnet surfaces in four-dimensional space forms Reviewed
Atsushi Fujioka
International Journal of Mathematics 15 ( 10 ) 981 - 985 2004
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Timelike Bonnet surfaces in Lorentzian space forms Reviewed
Atsushi Fujioka, Jun-ichi Inoguchi
Differential Geometry and its Applications 18 ( 1 ) 103 - 111 2003
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Spacelike surfaces with harmonic inverse mean curvature Reviewed
Atsushi Fujioka, Jun-ichi Inoguchi
The University of Tokyo, Journal of Mathematical Sciences 7 ( 4 ) 657 - 698 2000
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Surfaces with harmonic inverse mean curvature in space forms Reviewed
Atsushi Fujioka
Proceedings of the American Mathematical Society 127 ( 10 ) 3021 - 3025 1999
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On some generalisations of constant mean curvature surfaces Reviewed
Atsushi Fujioka, Jun-ichi Inoguchi
Lobachevskii Journal of Mathematics 3 73 - 95 1999
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Bonnet surfaces with constant curvature Reviewed
Atsushi Fujioka, Jun-ichi Inoguchi
Results in Mathematics 33 ( 3-4 ) 288 - 293 1998
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Actions of loop groups on simply connected H-surfaces in space forms Reviewed
Atsushi Fujioka
Journal of the Mathematical Society of Japan 50 ( 4 ) 819 - 829 1998
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Bonnet surfaces with constant curvature
1069 73 - 82 1998
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Harmonic maps and associated maps from simply connected Riemann surfaces into the 3-dimensional space forms Reviewed
Atsushi Fujioka
Tohoku Mathematical Journal 47 ( 3 ) 431 - 439 1995
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A generalization of H-surfaces and a certain duality Reviewed
Atsushi Fujioka
Journal of the Mathematical Society of Japan 47 ( 1 ) 183 - 190 1995
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Minimizing tangent maps from 3-ball to complex projective spaces Reviewed
Atsushi Fujioka
Journal of the Faculty of Science, University of Tokyo. Section IA 40 ( 1 ) 125 - 139 1993
Books
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曲線・曲面・多様体—現代幾何学の入り口
（ Role： Sole author）
日本評論社・大学数学ガイダンス 2024.4
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幾何学入門教室 : 線形代数から丁寧に学ぶ
藤岡, 敦（ Role： Sole author）
共立出版 2024.1 （ ISBN:9784320115552 ）
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手を動かしてまなぶ曲線と曲面
藤岡, 敦（ Role： Sole author）
裳華房 2023.9 （ ISBN:9784785315986 ）
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Basic Introduction to College Mathematics
FUJIOKA,Atsushi（ Role： Sole author）
2022.11 （ ISBN:9784320114821 ）
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手を動かしてまなぶε-δ論法
藤岡, 敦（ Role： Sole author）
裳華房 2022.8 （ ISBN:9784785315924 ）
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Advanced linear algebra through writing
FUJIOKA,Atsushi（ Role： Sole author）
2021.11 （ ISBN:9784785315917 ）
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Introduction to Information Geometry: Differential Geometry for Statistical Models
FUJIOKA,Atsushi（ Role： Sole author）
2021.4 （ ISBN:9784320114456 ）
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Set theory and general topology through writing
FUJIOKA,Atsushi（ Role： Sole author）
2020.8 （ ISBN:9784785315870 ）
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Calculus through writing
FUJIOKA,Atsushi（ Role： Sole author）
2019.8 （ ISBN:9784785315818 ）
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Manifolds through examples
FUJIOKA,Atsushi（ Role： Sole author）
2017.3 （ ISBN:9784785315719 ）
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数学で作るシャボン玉
藤岡敦（ Role： Sole author）
日本評論社・数学セミナー 2016.9
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Linear algebra through writing
FUJIOKA,Atsushi（ Role： Sole author）
2015.11 （ ISBN:9784785315641 ）
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４次元微分幾何学への招待
藤岡敦（ Role： Sole author）
サイエンス社・数理科学 2015.9
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よくわかる!基礎数学
藤田, 岳彦, 石村, 直之, 藤岡, 敦, 松田, 秀樹, 今井, 仁司, 石井, 昌宏, 竹内, 慎吾（ Role： Joint author）
実教出版 2012.11 （ ISBN:9784407325133 ）
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よくわかる微分積分
藤田, 岳彦, 石村, 直之, 藤岡, 敦, 松田, 秀樹, 今井, 仁司, 石井, 昌宏, 竹内, 慎吾（ Role： Joint author）
実教出版 2011.12 （ ISBN:9784407325119 ）
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よくわかる線形代数
藤田, 岳彦, 石村, 直之, 藤岡, 敦, 松田, 秀樹, 今井, 仁司, 石井, 昌宏, 竹内, 慎吾（ Role： Joint author）
実教出版 2011.12 （ ISBN:9784407325126 ）
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Primary 大学ノート 基礎数学
藤田, 岳彦, 石村, 直之, 藤岡, 敦（ Role： Joint author）
実教出版 2007.9 （ ISBN:9784407310801 ）
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Primary 大学ノート 線形代数
藤田, 岳彦, 石村, 直之, 藤岡, 敦（ Role： Joint author）
実教出版 2007.1 （ ISBN:9784407310818 ）
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Primary 大学ノート 微分積分
藤田, 岳彦, 石村, 直之, 藤岡, 敦（ Role： Joint author）
実教出版 2007.1 （ ISBN:9784407310825 ）
MISC
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Quadratic curves and equiaffine transformations
Atsushi Fujioka
Engineering & technology 20 1 - 6 2013.11
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Centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants
Atsushi Fujioka
Journal of Mathematical Analysis and Applications 365 ( 2 ) 694 - 700 2010.5
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Centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants
Atsushi Fujioka
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 365 ( 2 ) 694 - 700 2010.3
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Centroaffine Minimal Surfaces with Non-Semisimple Centroaffine Tchebychev Operator
Atsushi Fujioka
RESULTS IN MATHEMATICS 56 ( 1-4 ) 177 - 195 2009.12
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Geometry of the space of closed curves in the complex hyperbola
Atsushi Fujioka, Takashi Kurose
Kyushu Journal of Mathematics 63 ( 1 ) 161 - 165 2009.6
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GEOMETRY OF THE SPACE OF CLOSED CURVES IN THE COMPLEX HYPERBOLA
Atsushi Fujioka, Takashi Kurose
KYUSHU JOURNAL OF MATHEMATICS 63 ( 1 ) 161 - 165 2009.3
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Centroaffine minimal surfaces with non-semisimple centroaffine Tchebychev operator
Atsushi Fujioka
Results in Mathematics 56 ( 1 ) 177 - 195 2009.1
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Deformations of surfaces preserving conformal or similarity invariants
Atsushi Fujioka, Jun-Ichi Inoguchi
Progress in Mathematics 252 ( 252 ) 53 - 67 2007
-
Bonnet surfaces in four-dimensional space forms
Atsushi Fujioka
International Journal of Mathematics 15 ( 10 ) 981 - 985 2004.12
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Bonnet surfaces in four-dimensional space forms
Atsushi Fujioka
International Journal of Mathematics 15 ( 10 ) 981 - 985 2004
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Timelike Bonnet surfaces in Lorentzian space forms
Atsushi Fujioka, Jun-Ichi Inoguchi
Differential Geometry and its Application 18 ( 1 ) 103 - 111 2003.1
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"Timelike Bonnet surfaces in Lorentzian space forms," (jointly worked)
藤岡 敦(J. Inoguchi
Differential Geometry and its Applications 18 ( 1 ) 103 - 111 2003
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Actions of loop groups on simply connected H-surfaces in space forms
藤岡 敦
Journal of the Mathematical Society of Japan 50 no.4 819 - 829 1998
-
Actions of loop groups on simply connected H-surfaces in space forms
Atsushi Fujioka
Journal of the Mathematical Society of Japan 50 ( 4 ) 819 - 829 1998
-
A generalization of h-surfaces and a certain duality
Atsushi Fujioka
Journal of the Mathematical Society of Japan 47 ( 1 ) 183 - 190 1995
-
Harmonic maps and associated maps from simply connected Riemann surfaces into the 3-dimensional space forms
Atsushi Fujioka
Tohoku Mathematical Journal 47 ( 3 ) 431 - 439 1995
Presentations
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余等質性１の中心アファイン曲面
藤岡敦, 古畑仁
日本数学会２０２０年度年会（開催中止） 2020.3
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等積中心アファイン閉曲線のなす空間の間の同変射影 Invited
藤岡敦
Koriyama Geometry and Physics Days 2020 "Integrable systems, projective invariants, and related topics” 2020.2
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等積中心アファイン閉曲線のなす空間の間の同変射影
藤岡敦
淡路島幾何学研究集会2020 2020.1
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Centroaffine surfaces of cohomogeneity one Invited
FUJIOKA,Atsushi
2019.9
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回転面とアファイン微分幾何
藤岡敦
量子力学・確率論とその生命科学応用 2019.8
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余等質性１の中心アファイン曲面 Invited
藤岡敦
Workshop on Submanifold theory in a wider sense 2019.8
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等積中心アファイン平面閉曲線のなす空間と多重 Hamilton 構造
藤岡敦
微分同相群と平面閉曲線のなす空間 2019.2
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余等質性１の定曲率中心アファイン極小曲面
藤岡敦
淡路島幾何学研究集会2019 2019.1
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余等質性１の原点を中心とする固有アファイン球面
藤岡敦
淡路島幾何学研究集会2018 2018.1
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アファイン超曲面の中心写像
藤岡敦
幾何学コロキウム 2017.4
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Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection Invited
Atsushi Fujioka
The 2nd OCAMI-KOBE-WASEDA Joint International Workshop on Differential Geometry and Integrable Systems 2017.3
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中心アファイン曲面の３次形式
藤岡敦
淡路島幾何学研究集会2017 2017.1
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中心アファイン曲面に対する３次形式の平行性と再帰性 Invited
藤岡敦
２０１６年度福岡大学微分幾何研究集会 2016.11
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Tchebychev 作用素が半単純でない平坦中心アファイン曲面 Invited
藤岡敦
非可換幾何学と数理物理学 2015.7
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射影曲面と余次元２の前正規化された Blaschke はめ込み Invited
藤岡敦
RIMS 共同研究集会 部分多様体論と種々の幾何構造 2015.6
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射影曲面と余次元２の中心アファインはめ込み
藤岡敦
淡路島幾何学研究集会2015 2015.1
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射影曲面の持ち上げと簡約定理 Invited
藤岡敦
福岡大学微分幾何研究会 2014.11
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射影極小な中心アファイン極小曲面
藤岡敦
淡路島幾何学研究集会2014 2014.1
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射影および中心アファイン極小曲面
藤岡敦
大分幾何学セミナー 2013.9
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中心アファイン極小曲面と射影微分幾何
藤岡敦
幾何セミナー 2013.6
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等積中心アファイン曲線と多重 Hamilton 構造
藤岡敦
若狭三方幾何学研究集会2013 2013.3
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Differential geometry of projective or centroaffine surfaces Invited
藤岡敦
ミニワークショップ 統計多様体の幾何学とその周辺 (4) 2013.3
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Multi-Hamiltonian structures associated with the space of closed equicentroaffine curves Invited
Atsushi Fujioka
Workshop on Curve flows and integrable systems 2012.12
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等積中心アファイン平面閉曲線のなす空間に付随する多重 Hamilton 構造 Invited
藤岡敦
シンプレクティック幾何とその周辺2012 2012.11
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Centroaffine minimal surfaces Invited
Atsushi Fujioka
第１８回複素幾何シンポジウム 2012.10
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曲線の射影微分幾何
藤岡敦
OCU48セミナー 2012.5
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Equivariant projections between spaces of equicentroaffine curves Invited
Poisson geometry and related topics 23 2023.12
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等温漸近的な中心アファイン曲面
藤岡敦
Geometry Colloquium 2023.4
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定曲率中心 affine 極小曲面について
藤岡敦
Tsudoi KK 2022.3
-
Equivariant projections between spaces of closed equicentroaffine curves Invited
Atsushi Fujioka
2021.12
-
等積中心アファイン曲線と円周の微分同相群 Invited
藤岡敦
RIMS共同研究 部分多様体と関連する幾何構造研究の深化と融合 2021.6
Works
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千里山幾何学研究会
2023.6
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関大微分幾何研究会
2017.6
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関大微分幾何研究会
藤岡敦
2016.6
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関大微分幾何研究会
藤岡敦
2015.6
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関大微分幾何研究会
藤岡敦
2014.7
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Workshop on Curve flows and integrable systems
Atsushi Fujioka, Martin Guest
2012.12
Awards
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平成２７年度特別研究員等審査会専門委員（書面担当）表彰
2016.7 日本学術振興会
Research Projects
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曲線、曲面を巡るアファイン微分幾何と可積分系，射影微分幾何、情報幾何との交わり
Grant number：23K03101 2023.4 - 2027.3
日本学術振興会 科学研究費助成事業 基盤研究(C)
藤岡 敦
Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）
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Resonance of affine differential geometry and projective differential geometry from the viewpoint of integrability
Grant number：26400075 2014.4 - 2018.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Fujioka Atsushi
Grant amount：\4940000 （ Direct Cost: \3800000 、 Indirect Cost：\1140000 ）
We characterized indefinite projective surfaces developable to affine spheres which admit a lift to the affine space of codimension two satisfying a certain kind of the Einstein condition. We determined nondegenerate centroaffine surfaces whose cubic form is parallel or the traceless part of the cubic form is recurrent with respect to the induced connection. We determined nondegenerate centroaffine ruled surfaces satisfying geometric conditions that the center map is a point or a curve, or a part of the plane through the origin. We characterized proper affine spheres centered at the origin of cohomogeneity one, and classified centroaffine minimal surfaces with constant curvature of cohomegeneity one.
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Affine immersions in various fields
Grant number：26400058 2014.4 - 2018.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
FURUHATA Hitoshi, FUJIOKA Atsushi, HASEGAWA Izumi
Grant amount：\3900000 （ Direct Cost: \3000000 、 Indirect Cost：\900000 ）
We showed that a nondegenerate surface in the projective 3-space is locally constructed from a centroaffine minimal surface in the 4-dimensional vector space. The condition for such a surface to be projectively applicable to an affine sphere was obtained. We determined a nondegenerate centroaffine ruled surface such that the image of its center map lies on a plane containing the origin.
We defined the sectional curvature for a statistical structure, and found several properties of CR statistical submanifolds in a holomorphic statistical manifold of constant holomorphic sectional curvature, which are similar to well-known results for CR submanifolds in a Kaehler space form. We introduced notions of Sasakian statistical structure and Kenmotsu statistical structure, and gave their interesting examples. -
Development of Integrable Geometry
Grant number：23340012 2011.4 - 2015.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Miyaoka Reiko, KOTANI Motoko, NISHINOU Takeo, UEHARA Taketo, MATSUURA Nozomu, IWASAKI Katsunori, IRITANI Hiroshi, KAJIWARA Kenji, NAGATOMO Yasuyuki, NOMURA Takaaki, YAMADA Kotaro, ISHIKAWA Goo, UMEHARA Masaaki, GUEST Martin, SHODA Toshihiro, FUTAKI Akito, FUJIOKA Atsushi, RASSMAN Wayne, TAMARU Hiroshi
Grant amount：\13780000 （ Direct Cost: \10600000 、 Indirect Cost：\3180000 ）
Isoparametric hypersurfaces with 6 principal curvatures with multiplicity 2 are shown to be homogeneous, which solves one of Yau's problems. As for 4 principal curvature case, we gave a description by using the moment map of spin actions. Transnormal systems are investigated in details.
We show the non-existence of L2 harmonic 1-form on a complete non-compact stable minimal Lagrangian submanifolds in a Kaheler manifold with positive Ricci curvature. Then the number of non-parabolic ends is less than two, and in the surface case, the genus should vanish. The Floer theory on the intersection of a Lagrangian submanifold with its Hamiltonian deformation is investigated. The Gauss images of isoparametric hypersurfaces in the sphere are Lagrangian submanifolds of complex hyperquadric, and in this case, we show that if the multiplicities of the principal curvatures are bigger than 1, then they are Hamiltonian non-displaceable, -
Research on classical differential geometry from modern view points and its applications
Grant number：22540107 2010.4 - 2015.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
KUROSE Takashi, SUYAMA Yoshihiko, HAMADA Tatsuyoshi, KAWAKUBO Satoshi, MATSUURA Nozomu, INOGUCHI Junichi, FURUHATA Hitoshi, FUJIOKA Atsushi
Grant amount：\4160000 （ Direct Cost: \3200000 、 Indirect Cost：\960000 ）
In this research program, classical differential geometry, geometry of curves, surfaces and hypersurfaces in various spaces, have been studied, mainly with the method of the theory of integrable systems. Many results on classical differential geoemtry and its application have been achieved; for instance, through the observation that certain sorts of changes with time of curves yield equations dealt with in the theory of integrable systems, geometric descriptions and/or interpretations of several accomplishments of the theory have been given. Moreover, by applying geometry of hypersurfraces in affine spaces, new properties of statistical manifolds, which appear in informtion geometry, the study of mathematical statistics and information theory with differential geometric tools and methods, have been obtained and the statistical manifolds satisfying some curvature condition have been explicitely constructed and classified.
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Interaction of submanifold theory, integrable systems and variational method with a focus on affine differential geometry
Grant number：22540070 2010.4 - 2014.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
FUJIOKA Atsushi
Grant amount：\4160000 （ Direct Cost: \3200000 、 Indirect Cost：\960000 ）
We studied flat centroaffine surfaces with non-semisimple Tchebychev operator and showed that if the Pick invariant is constant then it is a centroaffine minimal surfaces described by the Meijer G-function. We also regarded centroaffine minimal surfaces, which is a natural generalization of proper affine spheres centered at the origin, as surfaces in the projective space and classified such surfaces whose associated surfaces are projective minimal.
We also studied surfaces in the complexified sphere parametrized by a complexified orthogonal net. Moreover, we introduced the countably infinite number of presymplectic structures on the space of closed equicentroaffine plane curves and showed that such a space can be described by multi-hamiltoninan system. -
Fusion of geometry and the theory of integrable systems
Grant number：19204006 2007 - 2010
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
MIYAOKA Reiko, OHNITA Yoshihiro, MOTOKO Kotani, SASAKI Takeshi, IWASAKI Katsunori, OYSU Yukio, KAJIWARA Kenji, NAGATOMO Yasuyuki, NAKAYAAHIKI Atsushi, YAMADA Kotaro, FUTAKI Akito, MARTIN Guest, WAYNE Rossman, SHODA Toshihiro, IRITANI Hiroshi, ISHIKAWA Goo, UMEHARA Masaaki, KAWAKUBO Satoshi, TAMARU Hiroshi, FUJIOKA Atsushi, MATSUURA Nozomu, NISHINOU Takeo
Grant amount：\27560000 （ Direct Cost: \21200000 、 Indirect Cost：\6360000 ）
We classified almost all isoparametric hypersurface, and characterize them in terms of the moment map, which proves the evidence of a relation with integrable systems. A basic theory of surfaces with singularities, and a new method using the Legendre map have been established. Via the Riemann-Hilbert correspondence, the dynamical system of Painleve equations is investigated, and the view point of the chaos has been developed. The modularity of higher genus Gromov-Witten and the mirror symmetry are discussed. A surface with potential appeared in quantum cohomology is constructed, which contributes to the tt* geometry.
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Research on stability and global properties of solutions of geometric variational problems
Grant number：19540217 2007 - 2009
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
KOISO Miyuki, FUJIOKA Atsushi, ANDO Naoya, PALMER Bennett, PICCIONE Paolo
Grant amount：\4550000 （ Direct Cost: \3500000 、 Indirect Cost：\1050000 ）
We studied equilibrium surfaces for anisotropic surface energy with volume constraint (CAMC surfaces). We obtained many important results about geometry and stability for CAMC surfaces with free boundary on two parallel planes. Moreover, we proved a uniqueness theorem for closed CAMC surfaces.
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General study on curves and surfaces related to integrable systems
Grant number：18540074 2006 - 2010
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
FUJIOKA Atsushi
Grant type：Competitive
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Research on global properties of solutions of geometric variational problems
Grant number：16540195 2004 - 2006
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
KOISO Miyuki, AIYAMA Reiko, FUJIOKA Atsushi
Grant amount：\2900000 （ Direct Cost: \2900000 ）
1. We studied critical points of an anisotropic surface energy for immersed surfaces in the euclidean three-space with a volume constraint. We assume that the surface energy satisfies a certain "convexity condition". In this case, the energy is called a constant coefficient elliptic parametric functional, and the critical points are surfaces with constant anisotropic mean curvature (CAMC surfaces). The energy-minimizer is a smooth convex surface which is called the Wulff shape (up to translation and homothety). In the case where the anisotropic surface energy is rotationally invariant, we obtained the following results.
(1) We proved that any embedded closed CAMC surface was (up to translation and homothety) the Wulff shape.
(2) We studied capillary surfaces for certain rotationally invariant elliptic parametric functionals supported on two horizontal planes separated by a fixed distance. When the wetting energy is nonnegative, we proved the existence and uniqueness of the stable solution. Also, we determined the geometric property of the solution. When the wetting energy is negative, we obtained some criterions for the existence and uniqueness of the stable solution. Also, we obtained some numerical results on this problem.
2. We obtained a new method of constructing examples of CAMC surfaces whose surface energy is not necessarily rotationally invariant. It will be useful not only for the study on CAMC surfaces but also for other fields such as crystallography, mathematical biology, and so on.
3. Anisotropic Delaunay surfaces are surfaces of revolution with constant anisotropic mean curvature. e showed how the generating curves of such surfaces could be obtained as the trace of a point attached to a curve which was rolled without slipping along a line. This generalizes the classical construction for constant mean curvature surfaces due to Delaunay. Moreover, we characterize anisotropic Delaunay surfaces by using their isothermic self-duality. -
曲面の微分幾何と可積分な微分方程式
Grant number：14740038 2002 - 2004
日本学術振興会 科学研究費助成事業 若手研究(B)
藤岡 敦
Grant amount：\2400000 （ Direct Cost: \2400000 ）
「第一基本形式と平均曲率では決まらない曲面を分類せよ」というBonnetによる問題は長い歴史をもつが、4次元空間形内における同様の考察は今まで殆どなされていない。そこで4次元空間形内の曲面で臍点のないものを考え、平均曲率ベクトルの長さを保ちながら局所的に等長的に変形できるものを3次元空間形内の場合に倣ってBonnet曲面とよぶことにすると、これは極小曲面や更に一般に平行な平均曲率ベクトルをもつ曲面を例として含むスペクトル径数をもつ曲面である。4次元空間形内のBonnet曲面のスペクトル径数は一般には2次特殊ユニタリ群に値をとるが、上の2つの例のように本質的に円周に値をとる場合を考え、このようなものを単純であるということにする。このとき、法ベクトル束が平坦な単純Bonnet曲面は全測地的または全臍的3次元空間形内のBonnet曲面であることが分かった。これは極小でない平行な平均曲率ベクトルをもつ曲面に対するChen-Yauの簡約定理の一般化とみなせる。Bonnet曲面のように点に依存するスペクトル径数をもつ曲面として他に調和逆平均曲率曲面とよばれる曲面が知られているが、Euclid空間内で考えるとこれらのもつ幾何的不変量の中に相似変換で不変なものが見い出される。そこで相似幾何的観点から両者を統一的に扱い、特徴的性質をもつものを分類した。また、共形幾何的観点からWillmore曲面がSchwarz微分を保つ変形をもつ曲面として特徴付けられることを示した。調和逆平均曲率曲面と同様な性質をもつ曲面としてEuclid空間内ではBianchi曲面とよばれる曲面が知られていたが、これは空間形内の場合にも自然に定義できることが分かった。更に調和逆平均曲率曲面の場合に知られていた、はめ込みをあたえる公式、曲面間の対応、特徴的な性質をもつものについて調べた。
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Functinal analytic approach to reseaches on deformation and stability of surfaces with constant mean curvature
Grant number：13640211 2001 - 2003
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
KOISO Miyuki, FUJIOKA Atsushi, AIYAMA Reiko, MIYAOKA Reiko
Grant amount：\3200000 （ Direct Cost: \3200000 ）
1.We studied the variational theory of surfaces whose mean curvature is prescribed to be a linear function of their height above a horizontal plane (PMC surfaces). We developed a flux formula and used it to prove nonexistence results for closed PMC surfaces. The perturbation theory for PMC surfaces was studied. We obtained necessary conditions for the stability of PMC surfaces with planar boundaries.
2.We posed a variational problem for surfaces whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour. The energy functional contains surface tension, a gravitational energy and a wetting energy, and the Euler-Lagrange equation can be expressed in terms of the mean curvature of the surface, the curvatures of the free boundary and a few other geometric quantities. Especially, we studied in detail a simple case where the solutions were vertical planar surfaces bounded by two vertical lines. We determined the stability or instability of each solution.
3.We studied the geometry of surfaces which are in equilibrium for an anisotropic surface energy with a volume constraint. We obtained the first and second variations and studied the exceptional set of the Gauss map for such surfaces. Also we obtained representation formulas of the equilibrium surfaces of revolution and studied the geometry and stability of these surfaces. -
曲面の微分幾何への可積分系理論的アプローチ
Grant number：12740037 2000 - 2001
日本学術振興会 科学研究費助成事業 奨励研究(A)
藤岡 敦
Grant amount：\1400000 （ Direct Cost: \1400000 ）
これまでの研究から続くような形で,可積分系理論的なアプローチから曲面の微分幾何についての研究を行った.3次元Euclid空間内の平均曲率一定曲面は,平均曲率が0でない場合,古くから可積分系理論の分野で知られているsine-Gordon方程式として記述される.平均曲率一定曲面の自然な一般化としてBonnet曲面(局所的に非自明に等長的に変形できる曲面)や調和逆平均曲率曲面(平均曲率の逆数が調和関数となる曲面)とよばれるものが定義される.特に,調和逆平均曲率曲面は,1994年にBobenkoによって3次元Euclid空間内の曲面の場合に定義された比較的新しい曲面族であり,Fokas-Gelfandによる特徴付けを経て,一般の3次元空間形内の曲面の場合に筆者により定義された.これらの曲面族は曲率線に沿った等温座標系が取れるとき,Christoffel変換とよばれる曲面の変換により互いに移り合うことが知られていた.今年度では,調和逆平均曲率曲面が曲率を用いて、表されるある量(3次元Euclid空間内の曲面の場合は主曲率の比)を保つような共形的変形をもつことに注目し,逆に調和逆平均曲率曲面をこのような量を保つ曲面の変換を許容するものとして特徴付けた.これは接線叢にある条件を加えた曲面同士の間の変換が存在するならばそれぞれの曲面のGauss曲率は負定数であるというBacklundの定理を想起させるが,調和逆平均曲率曲面に対してもより具体的に幾何的な特徴付けがされ得ることが期待できる.
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Homotopy theoretical research of manifolds
Grant number：11640069 1999 - 2001
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
ISHIMOTO Hiroyasu, TOMARI Masataka, MORISHITA Masanori, SUGANO Takashi, IWASE Zunici, FUJIOKA Atsushi
Grant amount：\2000000 （ Direct Cost: \2000000 ）
(1) Ishimoto studied the problem whether the matter corresponding to the Poincare conjecture holds or not for primary manifolds which are m-spheres with attached q-handles in the metastable range. For that purpose, he intended to extend the James-Whitehead theorem to the one for primary manifolds and succeeded in such case that the quadratic forms which distinguish primary manifolds take values in a cyclic group. Using the result, he proved in almost all cases that the matter in question also valid when the cyclic group is Z_<24>, adding to the results already obtained.
(2) Fujioka studied the fundamental properties of harmonic inverse mean curvature surfaces which are natural generalization of constant mean curvature surfaces. In particular, he characterized such a surface as the one which admits a transformation preserving a certain quantity represented with curvature. He studied also the Bonnet surfaces.
(3) Tomari studied the theory of multiplicity of filtered rings, and as an application, he constructed a criterion formula for the Milnor number of f which gives the definition of hyper surface isolated singularities, using the weight of coordinates and the Taylor expansion.
(4) Morishita studied analogies between knots and primes, 3-manifolds and number fields, basing on the analogy between link groups and Galois groups, and tried to bridge between the algebraic number theory and the 3-dimensional topology. He also studied with K, Murasugi in Toronto.
(5) Sugano studied the automorphic forms on unitary groups of degree 3 in number theory. He gave the explicit expansion for Eisenstein series and Kudla lift images using primitive theta functions, and gave the non-vanishing condition for the Kudla lift in terms of the periods. -
Study on automorphic forms, automorphic L functions and Shintani functions.
Grant number：11440005 1999 - 2001
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
SUGANO Takashi, FUJIOKA Atsushi, MORISHITA Masanori, ITO Tatsuro, MURASE Atsushi, HAYAKAWA Takayuki
Grant amount：\8300000 （ Direct Cost: \8300000 ）
1. Shintani functions (joint work with A. Murase and S. Kato)
We proved the multiplicity one property and explicit formulae of Whittaker-Shintani functions for split orthogonal groups.
2. Automorphic forms on unitary groups of degree 3 (joint work with A. Murase)
(1) We determined the refined Fourier-Jacobi expansion of holomorphic Eisenstein series and proved the algebraicity of Fourier-Jacobi coefficients.
(2) We determined the refinedd Fourier-Jacobi expansion of Kudla lift. We gave another proof of the Hecke compatibility of Kudla lift.
3. Construction of automorphic forms
(1) We constructed many automorphic forms on unitary groups of degree 3 explicity using Jacobi forms. Since their Taylor expansion can be writtten explictly, these forms will be used to determine the structure of the graded algebra.
(2) We started the study of Fourier expansion of Eisenstein series on 0(n+1,1) induced from automorphic forms on O(n). We hope that non-constant terms are described by using Shintani functions. -
共形不変形を持つ幾何学的変分問題の可積分系による解法について
Grant number：10640065 1998 - 2000
日本学術振興会 科学研究費助成事業 基盤研究(C)
落合 卓四郎, 藤岡 敦, 今野 宏
Grant amount：\3100000 （ Direct Cost: \3100000 ）
研究代表者の落合は、3次元ユークッリド空間内の閉じた曲面が共形不変性を持つ幾何学的変分問題(平成10年度に研究)の解であり、かつ群の作用を許す場合の研究を常微分方程式の分岐理論の立場から研究した。結果として、球面から変形してゆき、分岐する例を構成できた。
研究分担者の今野は、今年度は非可換群による超ケーラー商の研究を始めた。手始めとして、最も単純と思われる具体例を考察した。すなわち、複素直線の余接空間には、超ケーラー構造が入ることが知られているが、その直積のSO(3)による超ケーラー商のコホモロジー環を決定した。この空間は、複素直線の直積のSO(3)によるシンプレクテイック商(多角形のモジュライ空間とよばれる。)を半分次元の部分多様体として含んでいるが、可換群の場合と同様に、コホモロジー環は、超ケーラー商の方がより単純であった。さらに多角形のモジュライ空間はあるパラメータをもっていて、そのパラメータに応じてトポロジーが変化するが、超ケーラー商の部分空間として見ると、その変化の様子が大域的に記述される
研究分担者の藤岡はこれまでの研究から続くような形で,可積分系理論的なアプローチから曲面の微分幾何についての研究を行った.ユークリッド空間内の平均曲率一定曲面(CMC surfaceと略す)は,平均曲率が0でない場合,古くから可積分系理論の分野で知られているsine-Gordon方程式として記述される.ユークリッド空間内のCMC surfaceの自然な一般化としてBonnet曲面(局所的に非自明に等長的に変形できる曲面)やharmonic inverse mean curvature surface(平均曲率の逆数が調和関数となる曲面,以下,HIMC surfaceと略す)とよばれるものが定義される.これらの曲面族は曲率線に沿った等温座標系がとれる,isothermicとよばれる条件の下で,Hazzidakis方程式とよばれる常微分方程式によって記述されるが,この方程式は可積分系理論の分野で良く知られているPainleve方程式として記述されることが最近になってBobenko-Eitnerにより示された.Bonnet曲面やHIMC surfaceは定値計量とは限らない3次元空間形内の曲面に対しても定義される.そこで,不定値計量をもつ3次元空間形内の時間的なHIMC曲面について考察し,それに対するLax方程式やその解から曲面をあたえる公式である,immersion formulaを求めるといった,定値計量をもつ空間形内のHIMC surfaceと同様の性質について調べる他,時間的な曲面特有の現象についても調べた. -
可積分系の観点からみた平均曲率一定曲面の一般化
Grant number：10740028 1998 - 1999
日本学術振興会 科学研究費助成事業 奨励研究(A)
藤岡 敦
Grant amount：\1400000 （ Direct Cost: \1400000 ）
以前からの研究に引き続き、可積分系の観点からみた平均曲率一定曲面(CMCsurfaceと略す)の一般化について考察した.特に,不定値計量をもつ3次元空間形内の平均曲率を保ったまま,局所的に非自明,等長的に変形できる滑らかな曲面で,臍点をもたず計量が不定値であるもの(時間的Bonnet曲面とよばれる)について調べた.(時間的なCMCsurfaceで臍点をもたないものは時間的Bonnet曲面の例を与える.)このような曲面は大まかに分けて,局所的に,特定の曲率,及び,捩率をもつB-scrollとよばれるものと一対一に対応するか,または,generalized Hazzidakis equationとよばれる3階の常微分方程式に還元できるものであることが分かった.前者は時間的な曲面特有のものである.また,上述とは異なるCMCsurfaceの一般化としてユークリッド空間形内の曲面についてharmonic inverse mean curvature surface(HIMCsurfaceと略す)とよばれるものが定義されるのと同様に,不定値計量をもつ3次元空間形内の空間的なHIMC surfaceとよばれるものが定義されるが,このような局面でisothermicなものは上述のようにgeneralized Hazzidakis equationを解くことに帰着される.特にisothermicな曲面の例である回転面について考えると,軸が光的の場合は初等的に解ける解を与えることが分かった.この現象はユークリッド空間形内のHIMCsurfaceの場合には現れないものである.以上の結果については,現在,福岡大学の井ノ口順一氏と共著で投稿中である.
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Gemetric Structures on Manifolds and Global Analysis
Grant number：09440034 1997 - 1999
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
KOBAYASHI Osamu, FUJIOKA Atsushi, KITAHARA Haruo, KODAMA Akio, KATO Shin, KATAGIRI Minyo
Grant amount：\9200000 （ Direct Cost: \9200000 ）
Among many geometric structures of a manifold we are mainly interested in those structures which are closely related to the conformal geometry. Here are some of main results of this research project :
1. The scalar curvature equation. This equation describes the scalar curvature under a conformal change of a Riemannian metric. A systematic analysis has been done on non-compact manifolds, and the space of complete confomal metrics with prescribed scalar curvature is made clearer.
2. The Weyl structure. This is a torsion free affine connection that is compatible with a given conformal class. It is shown that the Ricci curvature is a complete invariant of a Weyl structure. Also conformally flat Einstein-Weyl structures on compact manifolds are classified.
3. Moebius geometry. The minimum number of vertices of a regular closed curve on the sphere with given topological type is completely determined in the case when the curve has at most five self-inter-sections. Also we introduce a Schwarzian derivative of a regular curve. This leads to new proofs of injectivity results of Nehari type. A gist is that a confomal strucutre of a manifold induces an integrable projective structure of a regular curve on the manifold. It is shown that injectivity of the projective development map of the curve implies the injectivity of the immersion to Moebius spaces. -
Automorphic forms and L functions on algebraic groups
Grant number：09640040 1997 - 1998
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
SUGANO Takashi, FUJIOKA Atsushi, HAYAKAWA Takayuki, MORISHITA Masanori, YAMADA Mieko, ITO Tatsuro
Grant amount：\3100000 （ Direct Cost: \3100000 ）
1. Automorphic L-functions on unitary groups (joint work with A.Murase)
We poved the analytic continuation and the functional equations of the standard L-function associated with a holomorphic cusp form on a unitary group. The key ingredient in the proof is a study of Shintani functions.
2. Automorphic forms on unitary groups of degree 3 and primitive theta functions (joint work with A.Murase)
For the study of arithmetical automorphic forms on a unitary group of degree 3, primitive theta functions are fundamental. We investigated a metaplectic representation of U(1) acting on the space of local primitive theta functions and gave its explicit irreducible decomposition.
As an application, we determined the refined Fourier-Jacobi expansion of holomorphic Eisenstein series.
3. theta lift and Jacobi group (joint work with A.Murase)
Since Kudla lift is a restriction of Oda lift, we can construct automorphic forms on unitary groups of degree 3 from Jacobi forms explicitly. We gave another formulation of refined Fourier-Jacobi expansion of holomorphic Eisenstein series using Jacobi type Eisenstein series.
Social Activities
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科学研究費委員会専門委員
2017.12 - 2020.11
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JP Journal of Geometry and Topology Editor
2016.5
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Zentralblatt MATH Reviewer
2016
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特別研究員等審査会専門委員（書面担当）
2014.8 - 2016.7
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Mathematical Reviews Reviewer
2014
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科学研究費委員会専門委員
2013.12 - 2014.11
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日本数学会 地方区代議員
2013.3 - 2014.2
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日本数学会 幾何学分科会拡大幹事
2010.10
Devising educational methods
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講義を主とする授業では、それまでに学んでんきたこと、あるいはこれから学ぶであろうことと関連付けながから講義を進めたほか、授業の後半では簡単な演習問題を出題し、その時間に扱った内容の確認をさせるとともに、学び合いの場も提供し、その結果を成績へ反映させた。
Teaching materials
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講義内容に関する資料をWeb上へ公開した。
Teaching method presentations
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特になし。
Special notes on other educational activities
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特になし。