Updated on 2024/09/20
博士(理学) ( 九州大学 )
理学修士 ( 九州大学 )
2018.4 -
東京工業大学(理学院)名誉教授
2018.4 -
中央大学理工学部教授
2016.4 - 2018.3
東京工業大学理学院(名称変更)・教授
2013.4 - 2016.3
東京工業大学大学院理工学研究科・教授
2010.4 - 2013.3
東北大学大学院情報科学研究科・教授
2004.4 - 2010.3
東京理科大学理工学部・教授
1995.10 - 2004.3
静岡大学理学部・准教授
1995.4 - 2004.3
静岡大学大学院理工学研究科担当
2001.4 - 2002.3
オレゴン大学数学教室・客員准教授
1991.4 - 1995.9
静岡大学教養部・助教授
1989.4 - 1991.3
日本文理大学工学部・講師
1987.4 - 1989.3
都城工業高等専門学校・講師
1986.4 - 1987.3
久留米工業高等専門学校一般科目・非常勤講師
1986.4 - 1987.3
東和大学一般科目・非常勤講師
日本数学会
Topology
Geometric Analysis
Differential Geometry
Natural Science / Geometry / 幾何学
The Yamabe problem on singular spheres and Obata-type theorems for csc metrics Invited
Reports on Geometry and Analysis in Fukuoka University 2019 97 - 105 2020
A gap theorem for positive Einstein metrics on the four-sphere Reviewed
K. Akutagawa, H. Endo, H. Seshadri
Math. Ann. 373 1329 - 1339 2019
The Yamabe problem on Dirichlet spaces Invited Reviewed
K. Akutagawa, G. Carron, R. Mazzeo
Tsinghua Lectures in Mathematics, Adv., Lect. in Math. 45 45 101 - 122 2018
Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space Reviewed
Reiko Aiyama, Kazuo Akutagawa, Satoru Imagawa, Yu Kawakami
ANNALI DI MATEMATICA PURA ED APPLICATA 196 ( 5 ) 1863 - 1875 2017.10
Proper harmonic maps between asymptotically hyperbolic manifolds Reviewed
Kazuo Akutagawa, Yoshihiko Matsumoto
MATHEMATISCHE ANNALEN 364 ( 3-4 ) 793 - 811 2016.4
Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group Reviewed
Reiko Aiyama, Kazuo Akutagawa
GEOMETRY AND TOPOLOGY OF MANIFOLDS 154 1 - 13 2016
Holder regularity of solutions for Schrodinger operators on stratified spaces Reviewed
Kazuo Akutagawa, Gilles Carron, Rafe Mazzeo
JOURNAL OF FUNCTIONAL ANALYSIS 269 ( 3 ) 815 - 840 2015.8
The Yamabe problem on stratified spaces Reviewed
Kazuo Akutagawa, Gilles Carron, Rafe Mazzeo
GEOMETRIC AND FUNCTIONAL ANALYSIS 24 ( 4 ) 1039 - 1079 2014.8
Semiumbilic points for minimal surfaces in Euclidean -space Reviewed
Reiko Aiyama, Kazuo Akutagawa
GEOMETRIAE DEDICATA 170 ( 1 ) 1 - 7 2014.6
SURFACES WITH INFLECTION POINTS IN EUCLIDEAN 4-SPACE Reviewed
Reiko Aiyama, Kazuo Akutagawa
KODAI MATHEMATICAL JOURNAL 37 ( 1 ) 174 - 186 2014.3
山辺不変量 Reviewed
芥川 和雄
数学 66 ( 1 ) 31 - 60 2014
Geometric relative Hardy inequalities and the discrete spectrum of Schrodinger operators on manifolds Reviewed
Kazuo Akutagawa, Hironori Kumura
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 48 ( 1-2 ) 67 - 88 2013.9
Biharmonic properly immersed submanifolds in Euclidean spaces Reviewed
Kazuo Akutagawa, Shun Maeta
GEOMETRIAE DEDICATA 164 ( 1 ) 351 - 355 2013.6
3-MANIFOLDS WITH POSITIVE FLAT CONFORMAL STRUCTURE Reviewed
Reiko Aiyama, Kazuo Akutagawa
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 140 ( 10 ) 3587 - 3592 2012.10
Computations of the orbifold Yamabe invariant Reviewed
Kazuo Akutagawa
MATHEMATISCHE ZEITSCHRIFT 271 ( 3-4 ) 611 - 625 2012.8
Aubin's Lemma for the Yamabe constants of infinite coverings and a positive mass theorem Reviewed
Kazuo Akutagawa
MATHEMATISCHE ANNALEN 352 ( 4 ) 829 - 864 2012.4
On Yamabe constants of Riemannian products Reviewed
Kazuo Akutagawa, Luis A. Florit, Jimmy Petean
COMMUNICATIONS IN ANALYSIS AND GEOMETRY 15 ( 5 ) 947 - 969 2007.12
Perelman's invariant, Ricci flow, and the Yamabe invariants of smooth manifolds Reviewed
Kazuo Akutagawa, Masashi Ishida, Claude LeBrun
ARCHIV DER MATHEMATIK 88 ( 1 ) 71 - 76 2007.1
3-Manifolds with yamabe invariant greater than that of RP3 Reviewed
Kazuo Akutagawa, André Neves
Journal of Differential Geometry 75 ( 3 ) 359 - 386 2007
The Yamabe invariants of orbifolds and cylindrical manifolds, and |rn|L^2-harmonic spinors Reviewed
Kazuo Akutagawa, Boris Botvinnik
Journal für die reine und angewandte Mathematik 574 121 - 146 2004.10
3次元多様体の山辺不変量 Reviewed
芥川和雄
21世紀の数学 --幾何学の未踏峰-- 309 - 334 2004.7
An obstruction to the positivity of relative Yamabe invariants Reviewed
Kazuo Akutagawa
Mathematische Zeitschrift 243 85 - 98 2003.4
Yamabe metrics on cylindrical manifolds Reviewed
K Akutagawa, B Botvinnik
GEOMETRIC AND FUNCTIONAL ANALYSIS 13 ( 2 ) 259 - 333 2003
The Weyl functional near the Yamabe invariant Reviewed
Kazuo Akutagawa, Boris Botvinnik, Osamu Kobayashi, Harish Seshadri
Journal of Geometric Analysis 13 ( 1 ) 1 - 20 2003
Manifolds of positive scalar curvature and conformal cobordism theory Reviewed
K Akutagawa, B Botvinnik
MATHEMATISCHE ANNALEN 324 ( 4 ) 817 - 840 2002.12
The relative Yamabe invariant Reviewed
K Akutagawa, B Botvinnik
COMMUNICATIONS IN ANALYSIS AND GEOMETRY 10 ( 5 ) 935 - 969 2002.12
The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space Reviewed
R Aiyama, K Akutagawa
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 14 ( 4 ) 399 - 428 2002.6
Notes on the relative Yamabe invariant Reviewed
Kazuo Akutagawa
Josai Mathematical Monographs 3 105 - 113 2001.2
Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space Reviewed
R Aiyama, K Akutagawa
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 52 ( 4 ) 877 - 898 2000.10
Kenmotsu type representation formula for spacelike surfaces in the de Sitter 3-space Reviewed
Reiko Aiyama, Kazuo Akutagawa
Tsukuba Journal of Mathematics 24 ( 1 ) 189 - 196 2000.6
A global correspondence between CMC-surfaces in S-3 and pairs of non-conformal harmonic maps into S-2 Reviewed
R Aiyama, K Akutagawa, R Miyaoka, M Umehara
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 128 ( 3 ) 939 - 941 2000.3
Representation formulas for surfaces in H-3(-c(2)) and harmonic maps arising from CMC surfaces Reviewed
R Aiyama, K Akutagawa
HARMONIC MORPHISMS, HARMONIC MAPS, AND RELATED TOPICS 413 275 - 285 2000
Minimal maps between the hyperbolic discs and generalized gauss maps of maximal surfaces in the anti-de sitter 3-space Reviewed
Reiko Aiyama, Kazuo Akutagawa, Tom Y. H. Wan
Tohoku Mathematical Journal 52 ( 3 ) 415 - 429 2000
Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere Reviewed
Reiko Aiyama, Reiko Aiyama
Tohoku Mathematical Journal 52 ( 1 ) 95 - 105 2000
Spin^c geometry, the Seiberg-Witten equations and Yamabe invariants of Kähler surfaces Reviewed
Kazuo Akutagawa
Interdisciplinary Information Sciences 5 55 - 72 1999.3
Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces in H-3(-c(2)) and S-1(3)(c(2)) Reviewed
R Aiyama, K Akutagawa
ANNALS OF GLOBAL ANALYSIS AND GEOMETRY 17 ( 1 ) 49 - 75 1999.2
Harmonic maps between hyperbolic spaces Reviewed
Kazuo Akutagawa
American Mathematical Society Translations, Sugaku Expositions 12 151 - 165 1999
Kenmotsu-Bryant type representation formula for constant mean curvature spacelike surfaces in H-1(3)(-c(2)) Reviewed
R Aiyama, K Akutagawa
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 9 ( 3 ) 251 - 272 1998.12
Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature Reviewed
K Akutagawa
PACIFIC JOURNAL OF MATHEMATICS 175 ( 2 ) 307 - 335 1996.10
双曲型空間の間の調和写像 Reviewed
芥川 和雄
数学 48 128 - 141 1996.8
Harmonic maps between unbounded convex polyhedra in hyperbolic spaces Reviewed
Kazuo Akutagawa, Seiki Nishikawa, Atsushi Tachikawa
Inventiones Mathematicae 115 ( 1 ) 391 - 404 1994.12
YAMABE METRICS OF POSITIVE SCALAR CURVATURE AND CONFORMALLY FLAT MANIFOLDS Reviewed
K AKUTAGAWA
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 4 ( 3 ) 239 - 258 1994.9
Harmonic diffeomorphisms of the hyperbolic plane Reviewed
Kazuo Akutagawa
Transactions of the American Mathematical Society 342 ( 1 ) 325 - 342 1994
Nonexistence results for harmonic maps between noncompact complete Riemannian manifolds Reviewed
Kazuo Akutagawa, Atsushi Tachikawa
Tokyo Journal of Mathematics 16 ( 1 ) 131 - 145 1993
Harmonic maps of unbounded convex polygons in the hyperbolic plane Reviewed
Kazuo Akutagawa, Seiki Nishikawa, Atsushi Tachikawa
Geometry and its Applications 17 - 19 1993
The Gauss map and spacelike surfaces with prescribed mean curvature in minkowski 3-space Reviewed
Kazuo Akutagawa, Seiki Nishikawa
Tohoku Mathematical Journal 42 ( 1 ) 67 - 82 1990
Compactness criteria for Riemannian manifolds with compact unstable minimal hypersurfaces Reviewed
Kazuo Akutagawa
Tsukuba Journal of Mathematics 13 ( 2 ) 505 - 512 1989.12
EXISTENCE OF MAXIMAL HYPERSURFACES IN AN ASYMPTOTICALLY ANTI-DESITTER SPACETIME SATISFYING A GLOBAL BARRIER CONDITION Reviewed
K AKUTAGAWA
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 41 ( 1 ) 161 - 172 1989.1
The Dirichlet problem at infinity for harmonic mappings between Hadamard manifolds Reviewed
Kazuo Akutagawa
Geometry of Manifolds (ed. by K. Shiohama) 59 - 70 1989
ON SPACELIKE HYPERSURFACES WITH CONSTANT MEAN-CURVATURE IN THE DE SITTER SPACE Reviewed
K AKUTAGAWA
MATHEMATISCHE ZEITSCHRIFT 196 ( 1 ) 13 - 19 1987
A note on spacelike hypersurfaces with prescribed mean curvature |rn|in a spatially closed globally static Lorentzian manifold Reviewed
Kazuo Akutagawa
Memoirs of the Faculty of Science, Kyūsyū University. Series A 40 119 - 125 1986.9
On a one-parameter subgroup of Ust(H) whose infinitesimal generator has only a singularly continuous part Reviewed
Kazuo Akutagawa
Memoirs of the Faculty of Science, Kyūsyū University. Series A 40 45 - 50 1986.2
Geometric Analysis
Kazuo Akutagawa（ Role： ContributorChapter 3:The Yamabe problem and the Yamabe invariant）
Asakura Publishing Co., Ltd. 2018.11
数学メモアール第7巻「山辺の問題」
小林治, 芥川和雄, 井関裕靖（ Role： Joint author10章 定理 C の証明 (3) |rn|12章 山辺不変量 |rn|参考文献 |rn|あとがき）
日本数学会 2013.12
岩波数学辞典第４版
彌永昌吉ほ, 芥川和雄を（ Role： Joint author8章6節 ：スカラー曲率と位相|rn|8章15節：非コンパクト多様体の間の調和写像）
岩波書店 2007.6
The Yamabe problem on singular spheres and an Obata-type theorem for csc metrics Invited
Kazuo Akutagawa
Differential Geometry Meeting of Fukuoka University 2019 2019.11
Obata-type Theorems on compact Einstein manifolds with boundary Invited
Kazuo Akutagawa
Math. Meeting of Riemannian Geometry and Geometric Analysis 2019.1
An Obata-type Theorem on compact Einstein manifolds with boundary Invited
Kazuo Akutagawa
Differential Geometry Meeting od Fukuoka University 2018 2018.11
Edge-cone Einstein metrics and the Yamabe invariant Invited
Kazuo Akutagawa
The 10th MSJ-SI: The Role of Metrics in the Theory of Partial Differential Equations 2018.7
A gap theorem for positive Einstein metrics on the four-sphere Invited
Kazuo Akutagawa
Geometry Seminar of Stanford University 2018.3
Edge-cone Einstein metrics and the Yamabe invariant Invited
Kazuo Akutagawa
The Joint Meeting of MSJ-TMS 2017.12
Edge-cone Einstein metrics and the Yamabe invariant Invited
Kazuo Akutagawa
Colloquium of Math. Department of Osaka University 2017.12
A gap theorem for positive Einstein metrics on the four-sphere Invited
Kazuo Akutagawa
Geometry Seminar of Math. Department of Osaka University 2017.12
Edge-cone Einstein metrics and the Yamabe invariant
Analysis, Geometry and Topology of Positive Scalar Curvature metrics 2017.8
A gap theorem for positive Einstein metrics on the four-sphere
Special Srssion "Differential Geometry" of the 3rd Congress of PRIMA 2017 2017.8
A gap theorem for positive Einstein metrics on the four-sphere
福岡大学理学部・福岡大学微分幾何セミナー 2017.6
Edge-cone Einstein 計量と山辺不変量
第63回幾何学シンポジウム 2016.8
Edge-cone Einstein metrics and the Yamabe invariant
Seminar on Geometry and Topology 2016.6
Edge-cone Einstein metrics and the Yamabe invariant
内藤博夫先生退官記念研究集会 2016.3
山辺の問題とその発展：特異空間上の山辺の問題とリッチフローとの関係
基礎自然科学系・学術セミナー 2016.3
Edge-cone Einstein metrics and the Yamabe invariant
京都大学大学院理学研究科・数学教室談話会 2015.12
山辺不変量と singular Einstein 計量--小林プログラムについて--
秋葉原微分幾何セミナー 2015.7
The Yamabe invariant and singular Einstein metrics
リーマン幾何と幾何解析 2015.3
The Yamabe invariant and singular Einstein metrics
東北大学大学院情報科学研究科数学教室・幾何解析セミナー 2015.1
Harmonic maps between asymptotically hyperbolic manifolds
東北大学大学院情報科学研究科数学教室・情報数理談話会 2015.1
多様体上の相対ハーディ型不等式とシュレーディンガー作用素の離散固有値
スペクトル・散乱理論とその周辺 (RIMS講究録) 2014.10
5次元ハイゼンベルグ群内の極小ルジャンドル曲面と正則データに よるワイエルシュトラス表現
部分多様体幾何とリー群作用 2014 2014.9
The Yamabe problem on Dirichlet spaces
確率論と幾何学 2014.9
Minimal Legendrian surfaces in the 5-dimensional Heisenberg group
Satellite Conference of Seoul ICM 2014, Geometric Analysis 2014.8
5次元ハイゼンベルグ群内の極小ルジャンドル曲面と正則データに よるワイエルシュトラス表現
福岡大学理学部・福岡大学微分幾何セミナー 2014.7
The Yamabe problem on edge-cone manifolds and Aubin's inequality
Stanford University Geometry Seminar 2014.3
特異空間上の山辺の問題，Aubin の不等式，edge-cone Einstein 計量
福岡大学理学部・福岡大学微分幾何セミナー 2013.12
The Yamabe problem on singular spaces and Dirichlet spaces
The 8th Pacific Rim Complex Geometry Conference 2013.8
The Yamabe problem on singular spaces
The Asian Mathematical Conference 2013 2013.7
特異空間および Dirichlet 空間上の山辺の問題
東京工業大学数学教室・大岡山談話会 2013.7
Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds
Variational Problems & Geometric PDE's 2013.6
特異空間上の山辺の問題
東北大学大学院情報科学研究科・情報数理談話会 2013.2
The Yamabe problem on stratified spaces
Tsinghua-Sanya Mathematical Forum 2013.1
On the existence of conic Yamabe metrics
仙台小研究会 --西川青季教授定年退職記念-- 2012.3
共形平坦な正の3次元多様体について
山口大学数理科学談話会 2012.2
3-manifolds with positive flat conformal structure
第10回環太平洋幾何学会議2011大阪-福岡 2011.12
Computations of the orbifold Yamabe invariant
Tambara Workshop on Parabolic Geometries and Related Topics I 2010.11
山辺不変量について --オービフォールドの山辺不変量--
2010年度日本数学会秋季総合分科会 (幾何学賞特別講演アブストラクト) 2010.9
Computations of the orbifold Yamabe invariant
Sacalar Curvature, Topology and Geometry 2010.8
オービフォールドの山辺不変量について
東北大学数学教室・談話会 2010.6
The Yamabe invariant of cylindrical manifolds and computations of the orbifold Yamabe invariant
5th Pacific Rim Conference on Mathematics 2010.6
完備多様体上のシュレーディンガー作用素の離散スペクトルについて
東北大学数学教室・幾何セミナー 2010.5
Notes on concordance and stable isotopy of positive scalar curvature
幾何と情報科学の架け橋 2010.3
山辺不変量について
東北大学大学院情報科学研究科数学教室・情報数理談話会 2009.7
The uncertainty principle lemma under gravity and the discrete spectrum of Schrodinger operators
ソボレフ不等式とその周辺 2009.3
The uncertainty principle lemma under gravity and the discrete spectrum of Schrodinger operators
大阪大学数学教室・幾何セミナー 2009.1
The uncertainty principle lemma under gravity and the discrete spectrum of Schrodinger operators
The 9th Pacific Rim Geometry Conference 2008.12
山辺不変量：手術理論と直積多様体に関する話題から
第55回幾何学シンポジウム 2008.8
Yamabe constants of infinite coverngs and a positive mass theorem
International Meeting on Spectral Geometry and Related Topics, Potsdam 2008 2008.5
On the Yamabe invariant of M x S^1
Mini-Workshop in Regensburg, Scalar curvature and semilinear PDEs in geometry and topology 2008.5
Perelman's invariant and the Yamabe invariant
TUS-NPU Bilatral Seminar 2008 (Proceedings) 2008.3
On the Yamabe invariant of M x S^1
The 3rd Geometry Friendship of Japan and Chaina 2008.1
On the Yamabe invariant of M x S^1
上智大学理工学部 数学教室談話会 2007.12
Yamabe constants of inifinite coverings and a positive mass theorem
Variational Problems in Geometry 2007.9
Yamabe constants of inifinite coverings and a positive mass theorem
International Coference in Geometry and Analysis, Nanjing 2007 2007.8
Perelman 不変量，Ricci flow および山辺不変量
リーマン幾何と幾何解析 2007.3
山辺不変量--共形幾何の広がり--
2007年度日本数学会年会・企画特別講演 (総合講演・企画特別講演アブストラクト) 2007.3
山辺の問題，Positive Mass Theorem および山辺不変量
特異点と時空，および関連する物理 2007.1
The Yamabe constants of infinite coverings and a positive mass theorem
Geometry of Singularities 2007.1
無限被覆空間の山辺定数と正質量定理
東京大学数理科学研究科・トポロジー火曜セミナー 2006.11
山辺不変量について
筑波大学数学系月例談話会 2006.9
山辺不変量と mass 不変量
東京幾何セミナー 2006.6
The mass of a compact positive conformal manifold
リーマン幾何と幾何解析 2006.2
逆平均曲率流と山辺不変量 Ｉ・ＩＩ
ENCOUNTER with MATHEMATICS --数学との遭遇-- 第35回 2005.12
低次元多様体の山辺不変量
東北大学大学院理学研究科・数学談話会 2005.10
正の山辺計量の判定とその応用(A criterion of positive Yamabe metrics and its applications)
第52回幾何学シンポジウム 2005.8
低次元多様体の山辺不変量--正の山辺計量の判定とその応用--
九州幾何セミナー 2005.7
3次元多様体の山辺不変量
日大トポロジーセミナー 2005.6
正質量定理
ENCOUNTER with MATHEMATICS --数学との遭遇-- 第32回 2005.1
リーマン的ペンローズ予想と逆平均曲率流
集中講義オーバービュー 2004.11
3次元多様体の山辺不変量と逆平均曲率流
集中講義オーバービュー 2004.11
山辺不変量と共形幾何(Yamabe Invariants and Conformal Geometry)
第51回幾何学シンポジウム 2004.8
Yamabe invariants of 3-manifolds
大域解析学とその周辺 2004.1
Yamabe invariants of 3-manifolds
東工大微分幾何国際研究集会 2003.12
ワイル汎関数と山辺不変量
筑波大幾何セミナー 2003.9
3次元多様体の山辺不変量，Total Mass に関する Penrose 予想および逆平均曲率流
第50回幾何学シンポジウム 2003.8
Yamabe metrics on cylindrical manifolds
The second International Symposium on Differential Geometry 2003.7
Decomposition along hypersurfaces and the Yamabe invariant of cylindrical manifolds
Stanford大学幾何セミナー 2003.3
ワイル汎関数と山辺不変量
東北大幾何セミナー 2003.1
The Yamabe invariants of cylindrical manifolds and orbifolds, and the Seiberg-Witten invariants
リー群と多様体の研究 2002.10
Yamabe metrics on cylindrical manifolds
Pacific Northwest Geometry/University of Oregon 2001.10
日本数学会・幾何学賞
2010.9 日本数学会・幾何学およびトポロジー分科会 山辺不変量の研究
Einstein metrics and Ricci flow on singular spaces, and study of the Yamabe invariant
Grant number：18H01117 2018.4 - 2023.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B) Chuo University
Grant amount： \16250000 （ Direct Cost: \12500000 、 Indirect Cost： \3750000 ）
Geometric and Global Analysis of Scalar Curvature and Einstein Metrics
Grant number：24340008 2012.4 - 2018.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
AKUTAGAWA KAZUO, FUTAKI Akito, KOBAYASHI Osamu, FURUTA Mikio, NAYATANI Shin, ONO Hajime, HONDA Shouhei, MATSUO Shinichiroh, MATSUMOTO Yoshihiko, Carron Gilles, Mazzeo Rafe, Mondello Ilaria, Vertman Boris
Grant amount： \17810000 （ Direct Cost: \13700000 、 Indirect Cost： \4110000 ）
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.
Geometric analysis for ideal boundaries of product manifolds, and the study of harmonic maps and Einstein metrics
Grant number：24654009 2012.4 - 2015.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
AKUTAGAWA Kazuo, KUMURA Hironori, AIYAMA Reiko, MAZZEO Rafe, MATSUMOTO Yoshihiko
Grant amount： \3640000 （ Direct Cost: \2800000 、 Indirect Cost： \840000 ）
On this research theme, I have obtained the following results: (1) the finiteness and infiniteness theorems of discrete spectrum of the Schrodinger operators on noncompact manifolds， (2) the existence and uniqueness theorems of the Dirichlet problem at infinity for harmonic maps between asymptotic hyperbolic manifolds， (3) the representational formula and halfspace theorem for minimal Legendrian surfaces in the 5-dimensional Heisenberg group， (4) the value distribution theorem for the Gauss maps of minimal Lagrangian surfaces in the complex 2-space.
Geometry of manifolds at infinity and the analytic property
Grant number：18540212 2006 - 2008
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
KUMURA Hironori, KASUE Atsushi, AKUTAGAWA Kazuo
Grant amount： \4000000 （ Direct Cost: \3400000 、 Indirect Cost： \600000 ）
GIT stability and canonical Kahler metrics
Grant number：18204003 2006 - 2008
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A) Tokyo Institute of Technology
FUTAKI Akito, MORITA Shigeyuki
Grant amount： \24570000 （ Direct Cost: \18900000 、 Indirect Cost： \5670000 ）
Study of Conformal Geometry and Group C^*-bundle from the Viewpoint of Global Analysis
Grant number：16540059 2004 - 2005
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Tokyo University of Science
AKUTAGAWA Kazuo, KOBAYASHI Osamu, MORIYOSHI Hitoshi, KUMURA Hironori, TONEGAWA Yoshihiro, IZEKI Hiroyasu
Grant amount： \3700000 （ Direct Cost: \3700000 ）
We have studied the following :
(1)Study of Yamabe Invariants
We estimated and determined the Yamabe invariant of some positive 3-manifolds, by using the inverse mean curvature flow and families of Green's functions. In especial, we classified completely all 3-manifolds with Yamabe invariant greater than that of RP^3. We also studied the positive Yamabe constants of Riemannian products and the behavior of them under magnifying one factor. We are now studying Aubin's type lemma for the positive Yamabe constants of infinite coverings, with some new results.
(2)Study of Conformal/Affine Geometry and group C^*-algebra
We gave new developments on conformal and projective geometry. In particular, we obtained an interesting varitional characterization of affine connections induced from Einstein metrics. We studied on twisted K-theory and groupoid C^*-algebras, and then proved a generalization of Gromov-Lawson theorem for foliated spaces.
(3)Study on Non-linear Analysis in Geometry
We studied on eigenvalue problem on complete manifolds, discrete groups and valiational problems on mean curvature. We then obtained results on non-existence of eigenvalues on complete manifolds of non-positive curvature, and a fixed-point theorem for discrete-group actions on Hadamard spaces.
Research on Jorgensen groups and Schottky spaces
Grant number：14540170 2002 - 2003
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
SATO Hiroki, AKUTAGAWA Kazuo, OKUMURA Yoshihide, NAKANISHI Toshihiro, OKUYAMA Yuusuke, KUMURA Hironori
Grant amount： \3300000 （ Direct Cost: \3300000 ）
We have studied the following four themes from 2002 to 2003. 1.Jorgensen groups. 2.The Picard group. 3.The Whitehead link group. 4.Classical Schottky spaces and Jorgensen number.
1.Jorgensen groups. A Jorgensen group is a non-elementary two-generator discrete group whose Jorgensen number is one. There are two types -parabolic type and elliptic type-for Jorgensen groups. Here we considered of parabolic type. There are three types for Jorgensen groups of parabolic type (finite type, countably infinite type and uncountably infinite type). We obtained the following. (1)We found all Jorgensen groups of finite type and all Jorgensen groups of countably infinite type in 2002, and (2)we found all Jorgensen groups of uncountably infinite type in 2003. Consequently we found all Jorgensen groups of parabolic type. The results (1) was talked at the International congress of Mathematicians in Beijing in 2002, and the result (2) was talked at Peking University in 2003.
2.The Picard group. We constructed a new fundamental region for the Picard group and we found eight relations for two generators of the group by using the fundamental region. This result was published in the Proceedings of the ISAAC Congress in Berlin in 2003.
3.The Whitehead link group. We proved that the Jorgensen number of the Whitehead link is two. Therefore the Whitehead link is not a Jorgensen group. We talked this result at the Internatonal Conference of Topology in 2002.
4.Classical Schottky spaces and Jorgensen number. We showed that there exists a classical Schottky group whose Jorgensen number is a given real number j【greater than or equal】4. We will talk this result at the International Conference of Potential Theory this summer.
Study on Geometry and Analysis of Conformal Manifolds and Bubbling Trees
Grant number：14540072 2002 - 2003
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
AKUTAGAWA Kazuo, OKUYAMA Yusuke, KUMURAKI Hironori, SATO Hiroki
Grant amount： \3600000 （ Direct Cost: \3600000 ）
We have studied the following:
1.Study of cylindrical and orbifold Yamabe invariants
As a generalization of the Yamabe constant/invariant of closed manifolds, we defined appropriately the orbifold Yamabe constant/invariant in terms of the cylindrical Yamabe constant/invariant.
For a cylindrical 4-manifold with positive cylindrical Yamabe invariant, we also established a method for estimating its cylindrical Yamabe invariant from above, by means of the Atiyah-Patodi-Singer L^2-index theory. Moreover, we generalized the Kobayashi inequality for Yamabe invariants to cylindrical Yamabe invariants, and studied its applications.
2.Study on the mass of compact conformal manifolds
The mass is a geometric invariant for asymptotically flat manifolds. For a compact conformal manifold (M, C) with positive Yamabe invariant, a scalar-flat, asymptotically flat manifold (M-{p},g_<AF>) is defined naturally from each initial metric g in C, where [g_<AF>]=C. Then the mass m(g ; p) is non-negative. This mass m(g ; p) also depends on the choice of g and p. However, if we use the Habermann-Jost's canonical metric g_<HJ> as a initial metric, then the mass m(g_<HJ>;p) is now independent of the choice of p. By using this fact, we can define the mass mass(M ; C) of the conformal manifold (M, C) as a conformal invariant. Moreover, taking the infimum of it over all conformal classes, we can also define the mass invariant mass(M) as a differential-topological invariant of M. We studied on the Kobayashi-type inequality of the mass invariant for connected manifolds.
3.Yamabe invariants of 3-manifolds
The method of inverse mean curvature flow is the central technique for the resolution of the Riemannian Penrose Conjecture in Cosmology. By using this technique, Bray-Neves determined the value of the Yamabe invariant of RP^3. This result is the first affirmative answer to the Schoen's Conjecture for the Yamabe invariant of 3-manifolds with constant curvature. We also determined the Yamabe invariant of the connected manifold RP^3 # k(S^2 x S^1), by means of the inverse mean curvature flow technique. This is also one of the open problems proposed by Bray-Neves.
For the above study, the support by the 'Grant-in-Aid for Sci. Res. (C)(2),14540072' was very important.
Analysis and Geometry of the Teichmuller Spaces
Grant number：13640164 2001 - 2002
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
OKUMURA Yoshihide, AKUTAGAWA Kazuo, NAKANISHI Toshihiro, SATO Hiroki, KUMURA Hisanori
Grant amount： \3400000 （ Direct Cost: \3400000 ）
My research during this term consists mainly of the following three branches :
1. Characterization of simple dividing loops on Riemann surfaces analytically.
2. Representation of the Teichmuller spaces global real analytically by angle parameters.
3. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.
I Characterized the geometry of Mobius transformations by using the one-half powers of these transformations and these traces. Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, by using the lifts of a Fuchsian group G representing S to the special linear group SL (2,C). For example, if S is a compact Riemann surface of genus p (>1), then the following is obtained :
The number of the lifts of G is 2 to the 2p-th power. Let g be an element of G corresponding to L. Then L is to be dividing if and only if for any lift of G, the matrix corresponding to g always has the negative trace.
I introduced new angle parameters corresponding to the intersection angles between geodesics on the marked Riemann surface, in order to obtain global real analytic and simple representations of the Teichmuller spaces. I showed that the Teichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces of types (1,1), (2,0) and (3,0). Angle parameters correspond to the intersection angles between the axes of the generators and these products of the marked Fuchsian group. I found out the high symmetry of the arrangement of these axes. I investigated the relation among such geometry of Mobius transformations, traces and angle parameters. From these observations, the much relation and information of angle parameters were obtained.
Next, I considered the representations of the Teichmuller modular groups by only angle parameters. I especially studied the following :
I. Interpretation of the Teichmuller modular groups as the actions of some special hyperbolic polygons bounded by the axes to others.
II. Relation between angle parameters and length parameters representing these groups.
Geometry of the Laplace operator
Grant number：13640069 2001 - 2002
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
KUMURA Hironori, AKUTAGAWA Kazuo, SATO Hiroki, KASUE Atsushi, OKUMURA Yoshihide
Grant amount： \3500000 （ Direct Cost: \3500000 ）
Kumura studied the relationship between analytic inequalities of noncompact Riemannian manifolds or compact Riemannian manifolds with boundary and its geometric information. To be concrete, he gave an intrinsic ultracontractive bound for compact Riemannian manifolds with nonconvex boundary, using their inner geometric property, by the arguments of Davies - Simon 1984. In order to do so, two inequalities, Hardy and Sobolev should be prepared. These inequalities are important. Indeed, for example, these induce an upper bound of the Neumann heat kernel, the boundary behavior of the Dirichlet heat kernel and Green kernel and the first gap of the Dirichlet eigenvalue. As for results on noncompact manifolds, the following results is obtained : generally, on noncompact Riemannian manifolds, the differential operator, Laplacian is defined, and its spectrum is closely related to the geometry of the manifolds and studied by many authors from various points of view. In particular, the essential spectrum of the Laplacian of noncompact complete Riemannian manifolds depends only on the geometry of the infinity of manifolds. Kumura considered the average of curvatures near the infinity with respect to some measure and studied its convergence and the essential spectrum of the Laplacian. He generalized a results of Donnelly and his own one.
Kasue studied the relationship between convergence of manifolds and Dirichlet forms, Sato studied the Jorgensen group, Akutagawa studied the Yamabe invariant and Okumura studied Teichmuller space from the global analytic viewpoint.
Research on Schottky spaces and Jorgensen groups
Grant number：12640168 2000 - 2001
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
SATO Hiroki, AKUTAGAWA Kazuo, OKUMURA Yoshihide, NAKANISHI Toshihiro, KUMURA Hironori
Grant amount： \3500000 （ Direct Cost: \3500000 ）
We have studied the following four themes from 2000 to 2001. 1. Jorgensen groups. 2. Jorgensen numbers of Classical Schottky spaces of real type. 3. The Picard group. 4. The Whitehead link.
1. J0rgensen groups. A Jorgensen group is a discrete group whose Jorgensen number is one. First we considered two one-parameter families. The results appeared in Contemporary Mathematics in 2000. Furthermore we studied Jorgensen groups of parabolic type. We talked the results at the Meeting of AMS (UCLA, 2000) and at the International Coference of Complex Analysis (China, 2000). Recently we found almost all Jorgensen groups of parabolic type. We talked about the results at the Meeting of Discontinuous Groups at Shizuoka University (January, 2002 and the Geometry and Topology Seminar at University of Oregon in March, 2002.
2. Jorgensen numbers of Classical Schottky space of real type. We found the best lower bounds for all kinds of the classical Schottky spaces of real type. The results appeared in J. Math. Soc. Japan in 2001.
3. The Picard group. We have appointed out before that the Picard group is a two-generator group. This time we constructed a new fundamental region for the group and we found eight relations by using the fundamental region. We talked this result at the ISAAC Congress in Berlin in 2001. The result will appear in the Proceedings.
4. The Whitehead link. We proved that the Jorgensen number of the Whitehead link is two. Therefore the Whitehead link is not a Jorgensen group. We talked the result at Kyoto University in 2001. We will talk this result at the internatonal conference in 2002.
TEICHMULLER SPACES AND GEOMETRY OF MOBIUS TRANSFORMATIONS
Grant number：11640162 1999 - 2000
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) SHIZUOKA UNIVERSITY
OKUMURA Yoshihide, AKUTAGAWA Kazuo, NAKANISHI Toshihiro, SATO Hiroki, KUMURA Hisanori
Grant amount： \3500000 （ Direct Cost: \3500000 ）
My research during this term consists mainly of the following three branches :
1. Consideration of the relation between angle parameters and the geometry of Mobius transformations.
2. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.
3. Characterization of simple dividing loops on Riemann surfaces analytically.
In order to obtain global real analytic and simple representations of the Teichmuller spaces, I introduced new angle parameters. I showed that the Theichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces.
Considering the axes of the generators and these products of Fuchsian groups, for example the once-holed torus Fuchsian groups, I found out the high symmetry of the arrangement of these axes. I investigated the relation among such geometry of Mobius transformations, traces and angle parameters, using the one-half powers of Mobius transformations and the hyperbolic geometry. From these observations, the much relation and information of angle parameters were obtained. From such information of angle parameters, I tried to represent the Teichmuller modular groups by only angle parameters. I considered the following :
(1) Relation between angle parameters and length parameters representing these groups.
(2) Concrete description of such groups by only angle parameters in the cases that it is easy to calculate.
(3) Choice of angle parameters (inductively) that simply represent the Theichmuller modular groups of the general cases.
I especially studied the representation of the Theichmuller modular groups of a once-holed torus and a compact Riemann surface of genus 2.
Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, using the lifts of a Fuchsian group G representing S to the special linear group SL (2, C). For example, if S is a compact Riemann surface of genus p (>1), then the following is obtained :
The number of the lifts of G is 2 to the 2p-th power. Let g be an element of G corresponding to L.Then L is to be dividing if and only if for any lift of G, the matrix corresponding to g always has the negative trace.
Spin^c Analysis for Group C^*-bundles on Manifolds and Study of Yamabe Invariants
Grant number：11640070 1999 - 2000
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
AKUTAGAWA Kazuo, KUMURA Hironori, OKUMURA Yoshihide, HIROKI Sato, IZEKI Hiroyasu, ONO Kaoru
Grant amount： \3400000 （ Direct Cost: \3400000 ）
We studied on Spin^c Analysis for Group C^*-bundles on Manifolds and Yamabe Invariants.
More precisely, we studied on the fundamentals of C^*-algebras and their K-theory. We also studied on the synthetic research of Yamabe invariants and Bordism theory with supergroups (i.e., supergroups, supergroup-structures on manifolds, supergroup C^*-bunbles and others). In particular, by using this bordism theory with supergroups, we obtained an obstruction to the positivity of relative Yamabe invariants. This result is written in the following paper :
Kazuo Akutagawa, An obstruction to the positivity of relative Yamabe invariants.
The head investigator, Kazuo Akutagawa also studied with professor Boris Botvinnik (University of Oregon) on Yamabe invariants from the algebraic topological viewpoint. Similar to the Homology theory, we defined a natural relative version of the Yamabe invariant, that is, the "relative Yamabe invariant", and we studied on it and the relation of the Yamabe invariant of a double manifold. This result is also written in the following papers :
Kazuo Akutagawa and Boris Botvinnik, Relative Yamabe invariant.
Kazuo Akutagawa and Boris Botvinnik, Manifolds of positive scalar curvature and conformal cobordism theory.
Discrete groups and geometry of ideal boundary
Grant number：11640056 1999 - 2000
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Tohoku University
IZEKI Hiroyasu, AKUTAGAWA Kazuo, NAKAGAWA Yasuhiro, SUNADA Toshikazu, NAYATANI Shin
Grant amount： \3200000 （ Direct Cost: \3200000 ）
The purpose of this project was to investigate the stability/rigidity of discrete groups from the viewpoint of geometry of the ideal boundary of negatively curved spaces and the cohomology of deiscrete groups. Our main result is summarized as follows.
Let Γ be a Kleinian group acting on n-sphere. If Γ is convex cocompact, the quotient of the domain of discontinuity is compact by definition. However, the converse is not true in general. Izeki (head investigator) showed that if the Hausdorff dimension of the limit set of Γ is less than n/2 and the quotient of the domain of discontinuity is compact, then Γ is convex cocompact. As a consequence, such a Γ is quasiconformally stable. We also gave several applications to topology and geometry of conformally flat manifolds with positive scalar curvature.
In case the Hausdorff dimension of the limit set is less than (n-2)/2, we found a proof using the index theorem for higher A^^<^>-genus. We applied the index theorem to the quotient of the domain of discontinuity. We note here what we mean by the ideal bounary is just the quotient of the domain of discontinuity. The higher A^^<^>-genus carries the information of the fundamental group, which turns out to be isomorphic to Γ in our case, and that is all that the higher A^^<^>-genus knows. And it is determined by the cohomology of Γ.
Research on Schottky groups and Schottky spaces
Grant number：10640158 1998 - 1999
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
SATO Hiroki, AKUTAGAWA Kazuo, NAKANISHI Toshihiro, OKUMURA Yoshihide, KUMURA Hironori
Grant amount： \3700000 （ Direct Cost: \3700000 ）
We have studied the following three themes from 1998 to 1999.
1. Classical Schottky space of real type.
2. Jorgensen numbers.
3. Jorgensen groups.
1. Classical Schottky space of real type. We classified the classical Schottky space of real type of genus two into eight types. We represented the shape of their spaces and found generators for the Schottky modular groups acting on the above spaces. Furthermore we determine fundamental regions for the Schottky modular groups.
2. Jorgensen numbers. We found the best lower bounds for all kinds of the classical Schottky spaces of real type considered in 1. We submited these results to J. Math. Soc. Japan.
3. Jorgensen groups. A Jorgensen group is a non-elementary discrete group whose Jorgensen number is one. In 1998 we considered two one-parameter families of Jorgensen groups. We talked about these results at State University of New York at Stony Brook, Rutgers University at Newark and at New Brunswick in 1998. The results has been printed in Contemporary Mathematics (the proceeding of The Second Ahlfors-Bers Colloquium) in February, 2000. In 1999 we considered distribution of Jorgensen groups on the fiber over the unit circle. In particular, we studied new two one-paramter families of Jorgensen groups on it. We talked about the results at the international conference (ISAAC) in August, 1999. We invited Professor I. Kra at State University of New York at Stony Brook to talk about Schottky groups and the Schottky space in January, 2000. We are preparing to submit some results obtained between April, 1999 and February, 2000.
Research for manifolds with conformal structure
Grant number：09440044 1997 - 1999
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B) FUKUOKA UNIVERSITY
SUYAMA Yoshihiko, KUROSE Takashi, AKUTAGAWA Kazuo, SHIOHAMA Katsuhiro, INOGUCHI Jun-ichi, YAMADA Kataro
Grant amount： \6600000 （ Direct Cost: \6600000 ）
1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere.
2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the conformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold.
3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form.
4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction.
Global Analysis for Geometric Structures and Topological Invariants
Grant number：09640102 1997 - 1998
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Shizuoka University
AKUTAGAWA Kazuo, HASHIMOTO Yoshitake, NAYATANI Shin, NAKANISHI Toshihiro, KUMURA Hironori, SATO Hiroki
Grant amount： \3000000 （ Direct Cost: \3000000 ）
We studied global analysis for geometric structures and topological invariants, as follows respectively.
Akutagawa : He studied on the theory of Seiberg-Witten theory on compact 4-manifolds, spin^c geome-try/analysis and its application to Yamabe invariants of K_hler surfaces. He obtained some strategies for open problems on Yamabe invariants.
Sato : He classified classical Schottky groups of real type of genus two into eight categories, and then obtained the fundamental domains of them and the shapes of their Schottky spaces.
Kumura : He studied on the spectral distance on compact Riemannian manifolds (M, g, upsilon) with weighted measure, by using their heat kernels. He also studied on compactness of a family of and the structure of the closure of {(M_i, g_i, upsilon_i)} with respect to the spectral distance. Moreover, he applied their results to some examples.
Nakanishi : He studied on the real analytic structure of the Teihm_ller spaces of 2-dimensional hyperbolic orbifolds of topologically finite. He realized the Teichm_ller spaces as real algebraic surfaces, and applied this result to the problems on the representation of mapping class groups, and the Weil-Petersson geometry.
Nayatani : He studied about the canonical metrics on the domains of discontinuity of discrete groups of complex-hyperbolic isometries. He defined quaternionic analogues of CR structures and pseudo-Hermitian sturctures paticularly, and applied them the study on the cannonical metrics.
Hashimoto : He studied on geometric structures induced by the Abelian differentials on Riemann surfaces. Moreover, from the viewpoint of the reduction of monodoromy of the projective structures on a Riemainn surface, he also gave a relation between the geometric structures and representation formulas of constant mean curvature surfaces.
Global Study of Conformal Riemannian Structures
Grant number：09640084 1997 - 1998
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Tohoku University
NAYATANI Shin, BANDO Shigetoshi, NISHIKAWA Seiki, NAKAGAWA Yasuhiro, IZEKI Hiroyasu, AKUTAGAWA Kazuo
Grant amount： \3100000 （ Direct Cost: \3100000 ）
Shin Nayatani considered the action of a discrete transformation group of a rank one symmetric space on its boundary at infinity, and studied a canonical invariant metric defined on the domain of discontinuity. In particular, he applied it to the study of the topology of the quotient manifold when the symmetric space is complex hyperbolic space, and also formulated the quaternionic analogue of CR structure/geometry, applying it to the study of the canonical metric in the quatenionic case. He also investigated geometric structures on the Furstenberg boundaries of some higher rank symmetric spaces.
Kazuo Akutagawa studied harmonic maps from hyperbolic space to a certain incomplete, negatively curved Riemannian manifold. In particular, he obtained results on the existence, uniqueness and regularity of solutions for the boundary value problem. He also made fundamental research on minimal maps between Riemannian manifolds, and obtained results on the existence and representation of minimal diffeomorphisms between hyperbolic disks.
Hiroyasu Izeki proved a vanishing theorem for the cohomology of flat Hilbert space bundles over a conformally flat manifold, obtained as the quotient of a spherical domain by a Kleinian group, and applied it to the study of the Hausdorff dimension of the limit set.
Yasuhiro Nakagawa investigated the Bando-Calabi-Futaki character, a generalization of the Futaki character. He extended Futaki-Morita's result that interpreted the Futaki character as a Godbillon-Vey invariant, to the case of the Bando-Calabi-Futaki character.
Seiki Nishikawa studied the boundary value problem for hamonic maps between homogenious Riemannian manifolds of negative curvature. In particular, he obtained results on the necessary condition which the boundary value should satisfy, the uniqueness of solution and the existence of solution for a suitable boundary value, in the case of the Carnot spaces.
Shigetoshi Bando studied the existence problem for Einstein metrics on Kahler manifolds and holomorphic vector bundles as well as its relation to the stability and degeneration phenomenon. He also investigated singular geometric structures which appeared as the limit of degeneration.
Global Analysis of Manifolds and Conformal Structures
Grant number：08304005 1996 - 1997
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A) UNIVERSITY OF TSUKUBA
ITOH Mitsuhiro, AKUTAGAWA Kazuo, TASAKI Hiroyuki, MABUCHI Toshiki, SEKIGAWA Kouei
Grant amount： \14700000 （ Direct Cost: \14700000 ）
The purposes of this research project was to accomplish synthetic investigation for geometry of manifolds, mainly for conformal structures and to obtain progressive reserach results.
By considering the last year's research progress situation we held in August, 1997 the geometry symposium, greatly organized, on a nationwide scale (participants 251, survey lectures 3, ordinary talks 53). In this symposium we could get exchanging of informations among researchers and intension and broadening of researches related to the purposes. As its consequences, for the research of global analysis on conformal structures M.Itoh, myself, the chief researcher, together with coresearchers, K.Akutagawa, S.Nayatani, Y.Izeki, S.Kato were able to obtain plenty of researches on the whole conformal geometry covering self-dual 4-manifolds, conformally flat manifolds and Einstein-Weyl manifolds.
In the other areas, that is, geometry of vector bundles, gauge fields great contributions were done by this project, especially in the field of self-dual bundles on quaternionic Kaehler manifolds. And also investigations done by this project took a progressive step in the direction of (almost) Kaehler structures. Furthermore, homogeneous manifolds, Rie-mannian geometry, symplectic geometry, geometry of curves and surfaces could get research progresses.
微分幾何学と大域解析学の研究
Grant number：08640100 1996
日本学術振興会 科学研究費助成事業 基盤研究(C) 静岡大学
芥川 一雄, 久村 裕憲, 横山 美佐子, 中西 敏浩, 板津 誠一, 佐藤 宏樹
Grant amount： \2300000 （ Direct Cost: \2300000 ）
当科学研究課題の目標は、(1)symplectic反自己双対4次元多様体の大域解析的研究、(2)3次元定曲率空間内の平均曲率一定曲面の研究であった。具体的研究成果は、下記の通りである。
(1)について:Seiberg-Witten理論全般、(LeBrun等による)そのリーマン幾何への応用(山辺不変量の評価、小平次元との関係)、およびsymplectic幾何(特に、Donaldsonのsymplectic部分多様体の理論)の基礎的研究(=学習)を行った。残念ながら今年度は、(1)については具体的研究成果が出せなかった。反自己双対共形構造は上記の幾何構造・理論と密接に関連しているものの、一方ではそれらと比較して研究の進んでいない対象でもある。今後の研究も、先ずは“symplectic構造付き"の反自己双対4次元多様体の研究を中心に進める予定である。
(2)について:相山氏(筑波大学)との共同研究において、R^3以外の3次元定曲率空間H^3,S^3内の平均曲率一定曲面、およびL^3以外の3次元ローレンツ定曲率空間S^3_1,H^3_1内の平均曲率一定な空間的曲面に対する、(調和写像をGauss写像とする)新しい表現公式を得た。応用として、“一般化されたGauss写像"と“2次的Gauss写像"の基本的関係も理解することができた。
(2)の研究成果は、次の論文にまとめており、現在投稿中または出版予定である。
[1]R.Aiyama and K.Akutagawa,Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces in H^3(-c^2) and S^3_1(c^2),preprint.
[2]R.Aiyama and K.Akutagawa,Kenmotsu-Bryant type representation formula for constant mean curvature surfaces in S^3(c^2),preprint.
[3]R.Aiyama and K.Akutagawa,Kenmotsu-Bryant type representation formula for constant mean curvature spacelike surfaces in H^3_1(-c^2),to appear in Differential Geometry and its Applications.
上記の研究(1)、(2)において、研究費補助金による研究連絡は極めて重要であった。
反自己双対計量の収束・退化の研究とそのトポロジーへの応用について
Grant number：07854002 1995
日本学術振興会 科学研究費助成事業 奨励研究(A) 静岡大学
芥川 一雄
Grant amount： \1100000 （ Direct Cost: \1100000 ）
本研究の目的は、コンパクト4次元多様体M上の反自己双対計量の次の族に、Gromov-Hausdorff距離を導入し、その列の収束・退化を調べることであった。【numerical formula】ただし、μ(M.[g])を共形類[g]の山辺不変量、μ_0を定数(正数とは限らない)とする。その際、山辺不変量の局所化から自然に定義されるSobolev
その際、山辺不変量の局所化から自然に定義されるSobolev 半径を物差しとしたMのthin-thick分解をその収束・退化の解析手段の基本とした。しかしながら次の2点が克服出来ず、本研究は不完全なものとなってしまった。
(1)Mの測地球の体積の上からの一様評価を証明すること(この評価はthick部分の計量のバブル現象の詳しい解析において必須である)。
(2)Mのthin部分にはF-構造がはいることを示す。
当該年度の前半において反自己双対計量の具体例の解析不足を認識し、後半は一貫してその具体例の持つ性質を調べた。その具体例とは、LeBrunのhyperbolic Ansatzおよびその一般化によるS^1-不変な反自己双対計量、JoyceによるT^2-不変な反自己双対計量等である。その過程で、知られている具体例は全て、局所的にスカラー平坦なケーラー計量に共形的であることに気づいた(複素構造も各点の近傍でのみ定義される)。その後、一般にそのような局所構造を持つかどうかを基本問題とし、反自己双対計量の局所的特徴付けの研究している。この研究方向は、今後の反自己双対計量の収束・退化の研究にも有用であると期待される。
上記の研究において、研究費補助金による研究連絡は極めて重要であった。
スカラー曲率が正の山辺計量の収束・退化の研究
Grant number：06854003 1994
日本学術振興会 科学研究費助成事業 奨励研究(A) 静岡大学
芥川 一雄
Grant amount： \900000 （ Direct Cost: \900000 ）
当科学研究課題の目的は、スカラー曲率が正の山辺計量の収束・退化の解析とその低次元多様体への応用であった。具体的研究成果は下記の通りである。
(1)山辺不変量正かつ体積1の山辺計量を持つn次元多様体の族Y(n)を考え、さらにある種の曲率積分有界の条件下で、収束・退化に関する成果を得た。これは、"11.研究発表の3番目の論文"の主結果の一般化である。
(2)山辺計量の族Y(n)を考え、(1)とは別のタイプの種々の曲率積分有界の条件下で、それらのコンパクト性定理や正の定曲率計量に対するピンチング定理を得た。特に新しいタイプの定理としては、3次元閉多様体上の平坦な共形構造に対するピンチング定理を得た。
以上(1)、(2)の研究成果は、次の論文にまとめており現在投稿中である。
"K.Akutagawa,Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature."
またこれらの研究過程において、スカラー曲率が正の山辺計量は、トポロジーへの応用上、3次元の場合が特に有用でありかつ幾つかの予想問題が自然に提出されることがわかった。
4次元の場合においても、反自己双対共形構造に対象を制限すれば、山辺計量の収束・退化の研究はそのモデュライ空間の研究に有用である(この場合には、山辺不変量の符号の条件は不必要となる)。実際"Sobolev半径"と言う概念が導入でき、それを物差しとして、反自己双対的山辺計量の収束・退化の解析はある程度可能であることもわかった。この対象においては具体例が豊富で、今後の研究は反自己双対的山辺計量の収束・退化の研究を中心に進める予定である。
上記の研究において、研究費補助金による研究連絡は極めて重要であった。
位相空間における関数空間・測度論とその関連分野の研究
Grant number：05640169 1993
日本学術振興会 科学研究費助成事業 一般研究(C) 静岡大学
大野 武, 芥川 一雄, 立川 篤, 根来 彬, 加藤 正公, 馬場 良和
Grant amount： \1200000 （ Direct Cost: \1200000 ）
1.Riesz空間Lで定義されたlocally solid Lebesgue topology7は、“如何なる条件のもとでL^u(Lのuniversally completion)上に拡張できるか"という問題はI.Labudaにより、種々検討され、1987年解決をみた。大野はそこで明らかにされた条件をさらに弱く設定し、Lがalmost σ-Dedekind完備である場合、L上のどのようなσ-Lebesgue topology 7がL^s(Lのσ-universally completion)上に拡張できるかを論究した。その結果、
(1) Lで定義されたRiesz σ-homomorphismはL^s上に一意に拡張できること。
(2) 7:locally convex-solid σ-Lebesgue topologyの場合、(1)と共に7がL^s上に拡張できるための必要十分条件を明らかにした。また、
(3) 7:locally solid σ-Lebesgue topologyの場合、7がL^sに拡張できるための必要十分条件は、Lで定義された任意のσ-Fatou topology μに対して、Lの非負・増加点列がμ-有界ならば、つねにそれは7-有界であることを明らかにした。さらに、
(4) L^sの表現問題およびLの双対空間による絶対弱位相について、有効な結果を得た。(投稿を予定している)
2. 関連分野の研究について、芥川は測地線による力学系、特に平坦な共形構造のモデュライ空間を山辺計量によって幾何学的に把握することから、そのコンパクト化および境界の詳しい解析にあたり、いくつかの優れた結果を得た。また、立川は芥川と共に差分近似法と変分法を用いて非線形偏微分方程式、特に非コンパクト完備リーマン体様体間の調和写像、および双曲型空間における非有界凸多面体間の調和写像の解析にあたり、有益な結果を得た。
3. 他の研究分担者もそれぞれの分野で良い結果が得られるよう、継続して研究を進めている。
幾何学における非線形問題の総合的研究
Grant number：05352001 1993
日本学術振興会 科学研究費助成事業 総合研究(B) 東北大学
西川 青季, 内藤 久資, 小磯 憲史, 芥川 和雄, 中島 啓, 板東 重稔
Grant amount： \2000000 （ Direct Cost: \2000000 ）
本研究は,多様体上の種々の幾何学的変分問題に対して,対象となる幾何構造を,各変分問題における停留点(良い幾何構造)へ変形していく過程に発生する特異点(幾何構造の退化)の構造とその発生のメカニズムを,対応する非線形偏微分方程式系の解の存在・収束・退化の観点から研究し,より統一的に解明することを目的として行われた。
そのために本研究では,研究分担者を組織委員として,この分野で最近著しい研究成果を挙げている研究者,とくに「幾何学と大域解析学」をテーマに開催された「第1回日本数学会国際研究集会」に招待講演者として選ばれた若手研究者を中心に,平成5年7月2日〜7月10日にかけて東北大学理学部においてワークショップを開催し,共同研究を行った。
この共同研究により,例えば,調和写像をはじめ弾性曲線やヤン・ミルズ場およびリーマン計量の共形変形に関する山辺の問題の研究などにおける,停留点への変形を記述する非線型発展方程式系の弱解の構成法が統一的に理解された。
この共同研究は,我が国におけるこの方面の共同研究をより緊密にするとともに,上記の国際研究集会において日本側の研究成果をより効果的に発表することにも大いに寄与し,その成果はこの国際研究集会のプロシーディングスに10篇の論文として発表された。
一方,この共同研究では,調和写像,極小曲面,ヤン・ミルズ場,アインシュタイン計量などの研究において幾何構造の退化として現れる,いわゆる「バブル・アップ」現象の解明にも焦点をあて,コンピューターシミュレーションによる模擬実験の現状について報告し合ったが,これについてはコンピューターグラフィクスの効率的利用法をはじめ,多くが今後の課題として残された。
2019.4 - 2020.3
The editorial office of Tokyo J. Math. The subchief of editorial committees of Tokyo J. Math.
2018.4 - 2020.3
The editorial office of Tokyo J. Math. an editorial committee of Tokyo J. Math.
2012.7 - 2018.6
Mathematical Society of Japan an editorial committee of J. Math. Soc. Japan
2013.4 - 2018.3
Tokyo Institute of Technology an editorial committee of Kodai Math. J.
2016.7 - 2017.6
Mathematical Society of Japan A committee of MSJ for Promoting Equal Participation of Men and Women in Science and Engineering
2014.11 - 2016.6
Mathematical Society of Japan A chief committee of MSJ for Promoting Equal Participation of Men and Women in Science and Engineering
2013.11 - 2014.10
Mathematical Society of Japan A committee of MSJ for Promoting Equal Participation of Men and Women in Science and Engineering
2012.4 - 2013.3
Mathematical Society of Japan a committee of Japanense journal Memoirs
2011.4 - 2013.3
Mathematical Society of Japan a borad of directors
2009.4 - 2013.3
Mathematical Society of Japan Geometry Section a secretary
2011.4 - 2012.3
Mathematical Society of Japan Geometry Section the chief committee
2010.4 - 2011.3
Mathematical Society of Japan Geometry Section a committee
2005.7 - 2007.6
Mathematical Society of Japan a committee of Japanese journal Sugaku
