1998 Fiscal Year Annual Research Report

調和写像に関係する幾何学

Research Project

Project/Area Number	09640092
Research Institution	TOKYO INSTITUTE OF TECHNOLOGY
Principal Investigator	宮岡礼子東京工業大学, 大学院・理工学研究科, 助教授 (70108182)
Co-Investigator(Kenkyū-buntansha)	厚地淳大阪大学, 大学院・理学研究科, 講師 (00221044) 小谷元子東邦大学, 理学部, 助教授 (50230024) 大仁田義裕都立大, 理学部, 教授 (90183764) 伊藤秀一東京工業大学, 大学院・理工学研究科, 助教授 (90159905) 二木昭人東京工業大学, 大学院・理工学研究科, 助教授 (90143247)
Keywords	調和写像 / 平均曲率一定曲面 / 極小曲面 / 等径超曲面 / 可積分系 / 値分布 / ゲージ理論 / サイバーグ・ウィッテン方程式
Research Abstract	本課題においては,調和写像論に関係して,曲面論(宮岡,小磯,佐藤,増田,成),超曲面論(宮岡),可積分系理論(宮岡,伊藤,大仁田,宇田川),格子やグラフ上の調和写像論(小谷),ゲージ理論的調和写像論(大仁田,橋口),リー群と対称空間(服部,田中),確率論的値分布論(厚地),サイバーグ-ウィッテン理論による微分位相不変量(二木),乗数イデアル層の応用(辻)の各主題に対して,主として次のような成果を得た. 宮岡は、非共形調和写像を極小曲面の成分として特徴付けた.また種数正のコンパクトリーマン面から5次元球面への非共形調和写像の大域変形の例を与え,この写像のnullityを下から評価した.次に3次元球面の平均曲率一定曲面と,2次元球面への非共形調和写像対との大域対応を示した.さらに主曲率の個数6の等径超曲面は等質なもの2種しかないことを,アイソスペクトラル原理と,Singerの定理を適用して示した.超曲面論に深く関わる流体型ハミルトン系の可積分性に関するFerapontovの仕事と,調和写像論から派生した,逆散乱法やヒエラルヒー,ベックルンド変換をポアッソン群作用を用いて説明するというTerng-Uhlenbeckの仕事を紹介した. 二木はケーラー・アインシュタイン計量や正のスカラー曲率を持つ計量の存在に関する障害を,サイバーグ・ウィッテン方程式から定義される微分位相幾何学的不変量を用いて研究した. 伊藤はハミルトン系以外の系(たとえば反転可能形[reversible system])の可積分性の条件について,特異点の存在する場合にLiouville-Arnoldの定理を拡張した. 大仁田はコンパクト・リーマン面からコンパクト対称空間への調和写像に付随したゲージ理論的方程式の解のモジュライ空間の構造を研究した.また,Hitchinのself-dua1ity equationsの解のhyperkaehlerモジュライ空間内のある複素ラグランジアン部分多様体の族を構成した. 小谷は結晶格子上の調和写像,熱核の漸近展開の無限時間での挙動,アルバネーゼ写像との関係等を研究した. 厚地はStochastic calculusの正則,調和写像の値分布への応用を調べ,Casorati-Weierstrass型の定理,Nevalinna有理形関数に対する対数微分の補題の一般化を行い,Cartan-Hadamard多様体の確率的完備極小部分多様体に、Omori,Jorge-Xavier等の結果を拡張した。

Research Products

(25 results)

All Other

All Publications (25 results)

[Publications] Atsushi Atsuji: "A Casorati-Weierstrass theorem for holomorphie mapray and invariant σ-fields of holomorphic diffusions" Bull.Sci.Math.to appear. (1999)
[Publications] Hidekazu Ito: "Integrability of syruple otic mappings and their Birkhoff normal forms" Tohoku Math.J.49. 73-114 (1997)
[Publications] 伊藤秀一: "ハミルトン力学系の最近の発展について" 物性研究. 70. (1998)
[Publications] Seiichi Udagawa: "An Exposition of McIntosh's work by showing many examples" 日本大学医学部一般教育研究紀要. 26. 13-35 (1998)
[Publications] Martin Guest and Yoshihiro Ohnita: "Actions of loop groups, deformations of harmonic maps, and their applications" American Math.Soc.Translation Ser.2. 183. 33-50 (1998)
[Publications] Miyuki Koiso: "Application of the maxmum principle to surfaces with constant mean curvature" 京都大学数理解析研究所講究録. 995. 1-6 (1998)
[Publications] Miyuki Koiso: "The stability and the vision number of surfaces with constant mean curvature" Bulltin of Kyoto Univ.of Education. B92. 1-11 (1998)
[Publications] Reiko Miyaoka: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces" Tohoku Math.J.51. 35-53 (1999)
[Publications] 宮岡礼子: "ポアッソン群作用と可積分系の散乱理論" 甲府報告集 Harmonic maps, Spectral curves and Soliton equations. 1-82 (1997)
[Publications] Reiko Miyaoka: "The splitting and deformations of the Gauss maps of compact CMC surfaces" Proceedings of Brest Workshop 1997 : Harmonic morphism, Harmonic maps and Related topics. (to appear). (1999)
[Publications] Reiko Miyaoka: "A global covrespondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2" Proceedings of DGA98. (to appear). (1999)
[Publications] Reiko Miyaoka: "Recent topics on harmonic map theory and related parts" Proceedings of the third international Workshop on Differential Geometry(1998), Kyungpook Univ.(to appea). (1999)
[Publications] Reiko Miyaoka: "On the theory of integrable systems and its applications" Proceedings of the theird International Workshops on Differential Geometry(1998), Kyungpook Univ.(to appear). (1999)
[Publications] Atsushi Atsuji: "A remark on a limitting behavior of the occupation times on unbounded domains of Brawnian motion" Jour.Math.Sci.Univ.Tokyo. (to appear). (1999)
[Publications] Miyuki Koiso: "The uniqueness for stable surfaces of constant mean curveture with free poindary" Bulletin of Kyoto Univ.of Education. B94. 1-7 (1999)
[Publications] Motoko Kotani: "Harmonic 2-spheres with γ pairs of extra eigenfunctions" Proc.Amer.Math.Soc.125. 2083-2092 (1997)
[Publications] Motoko Kotani: "Asymptotic behavior of the tranistion probability of a random walk on an infinite graple" J.Finct.Anal.159. 664-689 (1998)
[Publications] Toshikazu Sunad and Motoko Kotani: "Jacobian tori associated with a finite graph and its Abelian covering graph" Adv.in Apply.Math.(to appear). (1999)
[Publications] Weyne Rossmen and Katsunori Sato: "Constant mean curvature Surfaces with Two Ends in Hyperbolic Space" Experimental Math.7. (1998)
[Publications] Hajime Tsuji: "Existence and applications of analytic Zarishi decomposition" Proc.of 40-th Taniguchi Conference. (to appear).
[Publications] Hideko Hashiguchi: "The uniton unmbers of the harmonic 2-spheres in Gr_2(C^4)" J.Math.Soc.Japan. 49. 81-105 (1997)
[Publications] Tadashi Nagano and Makiko Tanaka: "The involutions of compact scpumetric spaces IV" Tokyo J.Math.(to appear).
[Publications] 二木昭人: "サイバーグ・ウィッテン不変量とトポロジー" 培風館, 196 (1998)
[Publications] 伊藤秀一: "常微分方程式と解析力学" 共立出版, 215 (1998)
[Publications] 二木昭人: "基礎講義線形代数学" 培風館, 176 (1998)

1998 Fiscal Year Annual Research Report

調和写像に関係する幾何学

Principal Investigator

宮岡 礼子 東京工業大学, 大学院・理工学研究科, 助教授 (70108182)

Research Products

[Publications] Atsushi Atsuji: "A Casorati-Weierstrass theorem for holomorphie mapray and invariant σ-fields of holomorphic diffusions" Bull.Sci.Math.to appear. (1999)

[Publications] Hidekazu Ito: "Integrability of syruple otic mappings and their Birkhoff normal forms" Tohoku Math.J.49. 73-114 (1997)

[Publications] 伊藤 秀一: "ハミルトン力学系の最近の発展について" 物性研究. 70. (1998)

[Publications] Seiichi Udagawa: "An Exposition of McIntosh's work by showing many examples" 日本大学医学部一般教育研究紀要. 26. 13-35 (1998)

[Publications] Martin Guest and Yoshihiro Ohnita: "Actions of loop groups, deformations of harmonic maps, and their applications" American Math.Soc.Translation Ser.2. 183. 33-50 (1998)

[Publications] Miyuki Koiso: "Application of the maxmum principle to surfaces with constant mean curvature" 京都大学数理解析研究所講究録. 995. 1-6 (1998)

[Publications] Miyuki Koiso: "The stability and the vision number of surfaces with constant mean curvature" Bulltin of Kyoto Univ.of Education. B92. 1-11 (1998)

[Publications] Reiko Miyaoka: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces" Tohoku Math.J.51. 35-53 (1999)

[Publications] 宮岡礼子: "ポアッソン群作用と可積分系の散乱理論" 甲府報告集 Harmonic maps, Spectral curves and Soliton equations. 1-82 (1997)

[Publications] Reiko Miyaoka: "The splitting and deformations of the Gauss maps of compact CMC surfaces" Proceedings of Brest Workshop 1997 : Harmonic morphism, Harmonic maps and Related topics. (to appear). (1999)

[Publications] Reiko Miyaoka: "A global covrespondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2" Proceedings of DGA98. (to appear). (1999)

[Publications] Reiko Miyaoka: "Recent topics on harmonic map theory and related parts" Proceedings of the third international Workshop on Differential Geometry(1998), Kyungpook Univ.(to appea). (1999)

[Publications] Reiko Miyaoka: "On the theory of integrable systems and its applications" Proceedings of the theird International Workshops on Differential Geometry(1998), Kyungpook Univ.(to appear). (1999)

[Publications] Atsushi Atsuji: "A remark on a limitting behavior of the occupation times on unbounded domains of Brawnian motion" Jour.Math.Sci.Univ.Tokyo. (to appear). (1999)

[Publications] Miyuki Koiso: "The uniqueness for stable surfaces of constant mean curveture with free poindary" Bulletin of Kyoto Univ.of Education. B94. 1-7 (1999)

[Publications] Motoko Kotani: "Harmonic 2-spheres with γ pairs of extra eigenfunctions" Proc.Amer.Math.Soc.125. 2083-2092 (1997)

[Publications] Motoko Kotani: "Asymptotic behavior of the tranistion probability of a random walk on an infinite graple" J.Finct.Anal.159. 664-689 (1998)

[Publications] Toshikazu Sunad and Motoko Kotani: "Jacobian tori associated with a finite graph and its Abelian covering graph" Adv.in Apply.Math.(to appear). (1999)

[Publications] Weyne Rossmen and Katsunori Sato: "Constant mean curvature Surfaces with Two Ends in Hyperbolic Space" Experimental Math.7. (1998)

[Publications] Hajime Tsuji: "Existence and applications of analytic Zarishi decomposition" Proc.of 40-th Taniguchi Conference. (to appear).

[Publications] Hideko Hashiguchi: "The uniton unmbers of the harmonic 2-spheres in Gr_2(C^4)" J.Math.Soc.Japan. 49. 81-105 (1997)

[Publications] Tadashi Nagano and Makiko Tanaka: "The involutions of compact scpumetric spaces IV" Tokyo J.Math.(to appear).

[Publications] 二木 昭人: "サイバーグ・ウィッテン不変量とトポロジー" 培風館, 196 (1998)

[Publications] 伊藤 秀一: "常微分方程式と解析力学" 共立出版, 215 (1998)

[Publications] 二木 昭人: "基礎講義線形代数学" 培風館, 176 (1998)

宮岡礼子東京工業大学, 大学院・理工学研究科, 助教授 (70108182)

[Publications] 伊藤秀一: "ハミルトン力学系の最近の発展について" 物性研究. 70. (1998)

[Publications] 二木昭人: "サイバーグ・ウィッテン不変量とトポロジー" 培風館, 196 (1998)

[Publications] 伊藤秀一: "常微分方程式と解析力学" 共立出版, 215 (1998)

[Publications] 二木昭人: "基礎講義線形代数学" 培風館, 176 (1998)