Updated on 2024/09/20

写真a

 
AKUTAGAWA Kazuo
 
Organization
Faculty of Science and Engineering Professor
Other responsible organization
Mathematics Course of Graduate School of Science and Engineering, Master's Program
Mathematics Course of Graduate School of Science and Engineering, Doctoral Program
Contact information
The inquiry by e-mail is 《here
External link

Degree

  • 博士(理学) ( 九州大学 )

  • 理学修士 ( 九州大学 )

Research History

  • 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

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Professional Memberships

  • 日本数学会

Research Interests

  • Topology

  • Geometric Analysis

  • Differential Geometry

Research Areas

  • Natural Science / Geometry  / 幾何学

Papers

  • 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

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  • A gap theorem for positive Einstein metrics on the four-sphere Reviewed

    K. Akutagawa, H. Endo, H. Seshadri

    Math. Ann.   373   1329 - 1339   2019

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  • 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

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  • 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

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    We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.

    DOI: 10.1007/s10231-017-0643-6

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  • Proper harmonic maps between asymptotically hyperbolic manifolds Reviewed

    Kazuo Akutagawa, Yoshihiko Matsumoto

    MATHEMATISCHE ANNALEN   364 ( 3-4 )   793 - 811   2016.4

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    Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with C-1 boundary conditions at infinity between asymptotically hyperbolic manifolds.

    DOI: 10.1007/s00208-015-1229-5

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  • Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group Reviewed

    Reiko Aiyama, Kazuo Akutagawa

    GEOMETRY AND TOPOLOGY OF MANIFOLDS   154   1 - 13   2016

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:SPRINGER JAPAN  

    In this paper, we give a representation formula for Legendrian surfaces in the 5-dimensional Heisenberg group h(5), in terms of spinors. For minimal Legendrian surfaces especially, such data are holomorphic. We can regard this formula as an analogue (in Contact Riemannian Geometry) of Weierstrass representation for minimal surfaces in R-3. Hence for minimal ones in h(5), there are many similar results to those for minimal surfaces in R3. In particular, we prove a Halfspace Theorem for properly immersed minimal Legendrian surfaces in h(5).

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

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:ACADEMIC PRESS INC ELSEVIER SCIENCE  

    We study the regularity properties for solutions of a class of Schrodinger equations (Delta + V)u = 0 on a stratified space M endowed with an iterated edge metric. The focus is on obtaining optimal Holder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential. (C) 2015 Elsevier Inc. All rights reserved.

    DOI: 10.1016/j.jfa.2015.02.003

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  • The Yamabe problem on stratified spaces Reviewed

    Kazuo Akutagawa, Gilles Carron, Rafe Mazzeo

    GEOMETRIC AND FUNCTIONAL ANALYSIS   24 ( 4 )   1039 - 1079   2014.8

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    We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik (GAFA 13:259-333, 2003) on the Yamabe problem on spaces with isolated conic singularities.

    DOI: 10.1007/s00039-014-0298-z

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  • Semiumbilic points for minimal surfaces in Euclidean -space Reviewed

    Reiko Aiyama, Kazuo Akutagawa

    GEOMETRIAE DEDICATA   170 ( 1 )   1 - 7   2014.6

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    For a minimal surface in the Euclidean 4-space, a semiumbilic point is at which its normal curvature vanishes. We prove that the semiumbilic points are isolated, if the normal curvature neither changes its sign nor vanishes identically. We also show that any entire minimal surface with flat normal bundle is an affine plane.

    DOI: 10.1007/s10711-013-9865-y

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  • SURFACES WITH INFLECTION POINTS IN EUCLIDEAN 4-SPACE Reviewed

    Reiko Aiyama, Kazuo Akutagawa

    KODAI MATHEMATICAL JOURNAL   37 ( 1 )   174 - 186   2014.3

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:KINOKUNIYA CO LTD  

    For a surface in the Euclidean 4-space, we prove a reduction theorem for the codimension of a surface all whose points are inflection points.

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  • 山辺不変量 Reviewed

    芥川 和雄

    数学   66 ( 1 )   31 - 60   2014

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    Language:Japanese   Publishing type:Research paper (scientific journal)   Publisher:日本数学会  

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  • 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

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    The classical Hardy inequality for the Laplacian Delta on shows the borderline-behavior of a potential V for the following question: whether the Schrodinger operator -Delta + V has a finite or infinite number of the discrete spectrum. In this paper, we will give a sharp generalization of this inequality on to a relative version of that on large classes of complete noncompact manifolds. Replacing by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some sharp criteria for the above-type question.

    DOI: 10.1007/s00526-012-0542-z

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  • Biharmonic properly immersed submanifolds in Euclidean spaces Reviewed

    Kazuo Akutagawa, Shun Maeta

    GEOMETRIAE DEDICATA   164 ( 1 )   351 - 355   2013.6

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    We consider a complete biharmonic immersed submanifold M in a Euclidean space . Assume that the immersion is proper, that is, the preimage of every compact set in is also compact in M. Then, we prove that M is minimal. It is considered as an affirmative answer to the global version of Chen's conjecture for biharmonic submanifolds.

    DOI: 10.1007/s10711-012-9778-1

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  • 3-MANIFOLDS WITH POSITIVE FLAT CONFORMAL STRUCTURE Reviewed

    Reiko Aiyama, Kazuo Akutagawa

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   140 ( 10 )   3587 - 3592   2012.10

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    In this paper, we consider a closed 3-manifold M with flat conformal structure C. We will prove that if the Yamabe constant of (M, C) is positive, then (M, C) is Kleinian.

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  • Computations of the orbifold Yamabe invariant Reviewed

    Kazuo Akutagawa

    MATHEMATISCHE ZEITSCHRIFT   271 ( 3-4 )   611 - 625   2012.8

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    We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order to prove it, we also solve the orbifold Yamabe problem under a certain condition. We use these results to give some exact computations of the Yamabe invariant of compact orbifolds.

    DOI: 10.1007/s00209-011-0880-0

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  • 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

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    Aubin's Lemma says that, if the Yamabe constant of a closed conformal manifold (M, C) is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. We generalize this lemma to the one for the Yamabe constant of any (M-infinity, C-infinity) of its infinite conformal coverings, provided that pi(1)(M) has a descending chain of finite index subgroups tending to pi(1)(M-infinity). Moreover, if the covering M-infinity is normal, the limit of the Yamabe constants of the finite conformal coverings (associated to the descending chain) is equal to that of (M-infinity, C-infinity). For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.

    DOI: 10.1007/s00208-011-0667-y

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  • 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

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    For a closed Riemannian manifold (M-m, g) of constant positive scalar curvature and any other closed Riemannian manifold (N-n, h), we show that the limit of the Yamabe constants of the Riemannian products (M x N, g + rh) as r goes to infinity is equal to the Yamabe constant of (M-m x R-n, [g + g(E)]) and is strictly less than the Yamabe invariant of Sm+n provided n >= 2. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:BIRKHAUSER VERLAG AG  

    In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals +infinity whenever the Yamabe invariant is positive.

    DOI: 10.1007/s00013-006-2181-0

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  • 3-Manifolds with yamabe invariant greater than that of RP3 Reviewed

    Kazuo Akutagawa, André Neves

    Journal of Differential Geometry   75 ( 3 )   359 - 386   2007

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:International Press  

    We show that, for all nonnegative integers k, l, m and n, the Yamabe invariant of #k(RP3)#ℓ(RP2 × S1)#m(S2 × S1)#n(S2 × S1) is equal to the Yamabe invariant of RP3, provided k + ℓ ≥ 1. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of RP3. More precisely, we show that such manifolds are either S3 or finite connected sums #m(S2 × S1)#n(S2 × S1), where S2 × S1 is the nonorientable S2-bundle over S1. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.© 2007 Applied Probability Trust. © 2007 Applied Probability Trust.

    DOI: 10.4310/jdg/1175266277

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  • 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

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  • 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

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  • An obstruction to the positivity of relative Yamabe invariants Reviewed

    Kazuo Akutagawa

    Mathematische Zeitschrift   243   85 - 98   2003.4

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  • Yamabe metrics on cylindrical manifolds Reviewed

    K Akutagawa, B Botvinnik

    GEOMETRIC AND FUNCTIONAL ANALYSIS   13 ( 2 )   259 - 333   2003

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    We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We find a positive solution to this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almost cuspsingularities. We describe the supremum case, i.e., when the cylindrical Yamabe constant is equal to the Yamabe invariant of the sphere. We prove that in this case such a cylindrical manifold coincides conformally with the standard sphere punctured at a finite number of points. In the course of studying the supremum case, we establish a Positive Mass Theorem for specific asymptotically flat manifolds with two almost conical singularities. As a by-product, we revisit known results on surgery and the Yamabe invariant.

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  • 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

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    For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant Y C(M) and the L n/2-norm W C(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y C(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant W C(M) is arbitrarily large. We study the image of the mapYW:C→(Y C(M), W C(M)) R 2 near the line {(Y(M), w)|w R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2. © 2003 Mathematica Josephina, Inc.

    DOI: 10.1007/BF02930992

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  • The relative Yamabe invariant Reviewed

    K Akutagawa, B Botvinnik

    COMMUNICATIONS IN ANALYSIS AND GEOMETRY   10 ( 5 )   935 - 969   2002.12

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    We define the relative Yamabe invariant of a compact smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the Yamabe invariant. We develop approximation techniques which lead to gluing theorems of two manifolds along their boundaries for the relative Yamabe invariant. We show that there are many examples of manifolds with both positive and non-positive relative Yamabe invariants. In particular, we construct families of four-manifolds with strictly negative relative Yamabe invariant and give an exact computation of the invariant.

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  • Manifolds of positive scalar curvature and conformal cobordism theory Reviewed

    K Akutagawa, B Botvinnik

    MATHEMATISCHE ANNALEN   324 ( 4 )   817 - 840   2002.12

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    For a supergoup gamma, we study closed gamma-manifolds with positive conformal classes. We use the relative Yamabe invariant from [2] to define the conformal cobordism relation on the category of such manifolds. We prove that the corresponding conformal cobordism groups Pos(*)(conf)(gamma) are isomorphic to the cobordism groups Pos(*)(gamma) defined by Stolz in [19]. As a corollary, we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics. Our main technical tools come from analysis and conformal geometry.

    DOI: 10.1007/s00208-002-0364-y

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  • 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

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    The purpose of this paper is to study some uniqueness, existence and regularity properties of the Dirichlet problem at infinity for proper harmonic maps from the hyperbolic m-space to the open unit n-ball with a specific incomplete metric. When m = n = 2, harmonic solutions of this Dirichlet problem yield complete constant mean curvature surfaces in the hyperbolic 3-space.

    DOI: 10.1007/s005260100109

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  • Notes on the relative Yamabe invariant Reviewed

    Kazuo Akutagawa

    Josai Mathematical Monographs   3   105 - 113   2001.2

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:城西大学  

    DOI: 10.20566/13447777_3_105

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:MATH SOC JAPAN  

    Our primary object of this paper is to give a representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space of curvature -1 in terms of their normal Gauss maps. For CMC (constant mean curvature) surfaces, we derive another representation formula in terms of their adjusted Gauss maps. These formulas are hyperbolic versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. As an application, we give a construction of complete simply connected CMC H (\H\ < 1) surfaces embedded in the hyperbolic 3-space.

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:筑波大学  

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  • 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

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    We show there is a global correspondence between branched constant mean curvature (i.e. CMC-) immersions in S-3/{+/-1} and pairs of non-conformal harmonic maps into S-2 in the same associated family. Furthermore, we give two applications.

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  • 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

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    Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:CHAPMAN & HALL/CRC PRESS  

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:東北大学  

    Problems related to minimal maps are studied. In particular, we prove an existence result for the Dirichlet problem at infinity for minimal diffeomorphisms between the hyperbolic discs. We also give a representation formula for a minimal diffeomorphism between the hyperbolic discs by means of the generalized Gauss map of a complete maximal surface in the anti-de Sitter 3-space. © 2000 Applied Probability Trust.

    DOI: 10.2748/tmj/1178207821

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:東北大学  

    Our primary object of this paper is to give a representation formula for a surface with prescribed mean curvature in the (metric) 3-sphere by means of a single component of the generalized Gauss map. For a CMC (constant mean curvature) surface, we derive another representation formula by means of the adjusted Gauss map. These formulas are spherical versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. Spin versions of them are obtained as well. © 2000 Applied Probability Trust.

    DOI: 10.2748/tmj/1178224660

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  • Spin^c geometry, the Seiberg-Witten equations and Yamabe invariants of Kähler surfaces Reviewed

    Kazuo Akutagawa

    Interdisciplinary Information Sciences   5   55 - 72   1999.3

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:東北大学  

    This is a survey article on spin/spinc geometry, the Seiberg-Witten equations and their applications to conformal Riemannian geometry, Yamabe invariants of Kähler surfaces particularly. The table of contents is the following:<BR>Section 1. Spin/Spinc Geometry<BR>Section 2. The Seiberg-Witten Equations<BR>Section 3. Yamabe Invariants of Kähler Surfaces

    DOI: 10.4036/iis.1999.55

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    Other Link: https://jlc.jst.go.jp/DN/JALC/00061158344?from=CiNii

  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:KLUWER ACADEMIC PUBL  

    Let c be a positive constant and H a constant satisfying \H\ &gt; c. Oar primary object of this paper is to give representation formulas for branched CMC H (constant mean curvature H) surfaces in the hyperbolic 3-space H-3(-c(2)) of constant curvature -c(2), and for spacelike CMC H surfaces in the de Sitter 3-space S-1(3)(c(2)) of constant curvature c(2). These formulas imply, for example, that every CMC H surface in H-3(-c(2)) can be represented locally by a harmonic map to the unit 2-sphere S-2.

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  • Harmonic maps between hyperbolic spaces Reviewed

    Kazuo Akutagawa

    American Mathematical Society Translations, Sugaku Expositions   12   151 - 165   1999

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:American Mathematical Society  

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:ELSEVIER SCIENCE BV  

    Our primary object of this paper is to give a representation formula for constant mean curvature spacelike surfaces (including maximal surfaces) in the anti-de Sitter 3-space H-1(3)(-c(2)) of constant negative curvature -c(2), Similar to the Kenmotsu type representation formula for nonzero constant mean curvature spacelike surfaces in the Minkowski S-space L-3. This formula implies that every simply connected spacelike surface M with constant mean curvature in H-1(3)(-c(2)) can be represented by a harmonic map from M to the hyperbolic 2-plane H-2.

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  • 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

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:PACIFIC JOURNAL MATHEMATICS  

    Let Y-1(n, mu(0)) be the class of compact connected smooth n-manifolds M (n greater than or equal to 3) with Yamabe metrics g of unit volume which satisfy
    mu(M, [g]) greater than or equal to mu(0) &gt; 0,
    where [g] and mu(M, [g]) denote the conformal class of g and the Yamabe invariant of (M, [g]), respectively. The purpose of this paper is to prove several convergence theorems for compact Riemannian manifolds in Y-1(n, mu(0)) with integral bounds on curvature. In particular, we present a pinching theorem for hat conformal structures of positive Yamabe invariant on compact 3-manifolds.

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  • 双曲型空間の間の調和写像 Reviewed

    芥川 和雄

    数学   48   128 - 141   1996.8

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  • Harmonic maps between unbounded convex polyhedra in hyperbolic spaces Reviewed

    Kazuo Akutagawa, Seiki Nishikawa, Atsushi Tachikawa

    Inventiones Mathematicae   115 ( 1 )   391 - 404   1994.12

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    DOI: 10.1007/BF01231765

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  • YAMABE METRICS OF POSITIVE SCALAR CURVATURE AND CONFORMALLY FLAT MANIFOLDS Reviewed

    K AKUTAGAWA

    DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS   4 ( 3 )   239 - 258   1994.9

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    Let CY(n, mu0, R0) be the class of compact connected smooth manifolds M of dimension n greater-than-or-equal-to 3 and with Yamabe metrics g of unit volume such that each (M, g) is conformally flat and satisfies
    mu(M,[g]) greater-than-or-equal-to mu0 &gt; 0, integral(M)E(g)\n/2dupsilon(g) less-than-or-equal-to R0,
    where [g], mu(M,[g]) and E(g) denote the conformal class of g, the Yamabe invariant of (M,[g]) and the traceless part of the Ricci tensor of g, respectively. In this paper, we study the boundary partial-derivativeCY(n,mu0, R0) of CY(n, mu0, R0) in the space of all compact metric spaces equipped with the Hausdorff distance. We shall show that an element in partial-derivativeCY(n, mu0, R0) is a compact metric space (X, d). In particular, if (X, d) is not a point, then it has a structure of smooth manifold outside a finite subset S, and moreover, on X/S there is a conformally flat metric g of positive constant scalar curvature which is compatible with the distance d.

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  • Harmonic diffeomorphisms of the hyperbolic plane Reviewed

    Kazuo Akutagawa

    Transactions of the American Mathematical Society   342 ( 1 )   325 - 342   1994

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    In this paper, we consider the Dirichlet problem at infinity for harmonic maps between the Poincaré model D of the hyperbolic plane H2, and solve this when given boundary data are C4immersions of D(∞), the boundary at infinity of D, to D(∞). Also, we present a construction of nonconformal harmonic diffeomorphisms of D, and give a complete description of the boundary behavior, including their first derivatives. © 1994 American Mathematical Society.

    DOI: 10.1090/S0002-9947-1994-1147398-9

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  • Nonexistence results for harmonic maps between noncompact complete Riemannian manifolds Reviewed

    Kazuo Akutagawa, Atsushi Tachikawa

    Tokyo Journal of Mathematics   16 ( 1 )   131 - 145   1993

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    DOI: 10.3836/tjm/1270128986

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  • Harmonic maps of unbounded convex polygons in the hyperbolic plane Reviewed

    Kazuo Akutagawa, Seiki Nishikawa, Atsushi Tachikawa

    Geometry and its Applications   17 - 19   1993

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  • 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

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    DOI: 10.2748/tmj/1178227694

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  • Compactness criteria for Riemannian manifolds with compact unstable minimal hypersurfaces Reviewed

    Kazuo Akutagawa

    Tsukuba Journal of Mathematics   13 ( 2 )   505 - 512   1989.12

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  • 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

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  • The Dirichlet problem at infinity for harmonic mappings between Hadamard manifolds Reviewed

    Kazuo Akutagawa

    Geometry of Manifolds (ed. by K. Shiohama)   59 - 70   1989

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  • ON SPACELIKE HYPERSURFACES WITH CONSTANT MEAN-CURVATURE IN THE DE SITTER SPACE Reviewed

    K AKUTAGAWA

    MATHEMATISCHE ZEITSCHRIFT   196 ( 1 )   13 - 19   1987

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  • 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

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  • 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

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▼display all

Books

  • Geometric Analysis

    Kazuo Akutagawa( Role: ContributorChapter 3:The Yamabe problem and the Yamabe invariant)

    Asakura Publishing Co., Ltd.  2018.11 

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    Total pages:436pages   Language:Japanese   Book type:Dictionary, encyclopedia

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  • 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 

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    Responsible for pages:44-61,64-69,70-74,75   Language:Japanese   Book type:Scholarly book

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  • 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 

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    Responsible for pages:788-789,985-985   Language:Japanese  

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Presentations

  • 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 

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  • Obata-type Theorems on compact Einstein manifolds with boundary Invited

    Kazuo Akutagawa

    Math. Meeting of Riemannian Geometry and Geometric Analysis  2019.1 

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    Event date: 2019.1    

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  • An Obata-type Theorem on compact Einstein manifolds with boundary Invited

    Kazuo Akutagawa

    Differential Geometry Meeting od Fukuoka University 2018  2018.11 

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  • 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 

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  • A gap theorem for positive Einstein metrics on the four-sphere Invited

    Kazuo Akutagawa

    Geometry Seminar of Stanford University  2018.3 

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  • Edge-cone Einstein metrics and the Yamabe invariant Invited

    Kazuo Akutagawa

    The Joint Meeting of MSJ-TMS  2017.12 

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  • Edge-cone Einstein metrics and the Yamabe invariant Invited

    Kazuo Akutagawa

    Colloquium of Math. Department of Osaka University  2017.12 

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  • A gap theorem for positive Einstein metrics on the four-sphere Invited

    Kazuo Akutagawa

    Geometry Seminar of Math. Department of Osaka University  2017.12 

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  • Edge-cone Einstein metrics and the Yamabe invariant

    Analysis, Geometry and Topology of Positive Scalar Curvature metrics  2017.8 

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  • A gap theorem for positive Einstein metrics on the four-sphere

    Special Srssion "Differential Geometry" of the 3rd Congress of PRIMA 2017  2017.8 

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  • 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 

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  • 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 

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  • Edge-cone Einstein metrics and the Yamabe invariant

    Seminar on Geometry and Topology  2016.6 

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  • Edge-cone Einstein metrics and the Yamabe invariant

    内藤博夫先生退官記念研究集会  2016.3 

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  • 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 

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  • 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 

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  • 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 

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  • The Yamabe invariant and singular Einstein metrics

    リーマン幾何と幾何解析  2015.3 

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  • 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 

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  • 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 

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  • 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 

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  • 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 

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  • The Yamabe problem on Dirichlet spaces

    確率論と幾何学  2014.9 

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  • Minimal Legendrian surfaces in the 5-dimensional Heisenberg group

    Satellite Conference of Seoul ICM 2014, Geometric Analysis  2014.8 

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  • 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 

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  • The Yamabe problem on edge-cone manifolds and Aubin's inequality

    Stanford University Geometry Seminar  2014.3 

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  • 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 

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  • The Yamabe problem on singular spaces and Dirichlet spaces

    The 8th Pacific Rim Complex Geometry Conference  2013.8 

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  • The Yamabe problem on singular spaces

    The Asian Mathematical Conference 2013  2013.7 

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  • 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 

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  • Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds

    Variational Problems & Geometric PDE's  2013.6 

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  • 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 

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  • The Yamabe problem on stratified spaces

    Tsinghua-Sanya Mathematical Forum  2013.1 

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  • 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 

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  • 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 

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  • 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 

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  • Computations of the orbifold Yamabe invariant

    Tambara Workshop on Parabolic Geometries and Related Topics I  2010.11 

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  • 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 

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  • Computations of the orbifold Yamabe invariant

    Sacalar Curvature, Topology and Geometry  2010.8 

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  • 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 

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  • The Yamabe invariant of cylindrical manifolds and computations of the orbifold Yamabe invariant

    5th Pacific Rim Conference on Mathematics  2010.6 

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  • 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 

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  • Notes on concordance and stable isotopy of positive scalar curvature

    幾何と情報科学の架け橋  2010.3 

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  • 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 

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  • 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 

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  • 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 

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  • The uncertainty principle lemma under gravity and the discrete spectrum of Schrodinger operators

    The 9th Pacific Rim Geometry Conference  2008.12 

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  • 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 

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  • Yamabe constants of infinite coverngs and a positive mass theorem

    International Meeting on Spectral Geometry and Related Topics, Potsdam 2008  2008.5 

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  • On the Yamabe invariant of M x S^1

    Mini-Workshop in Regensburg, Scalar curvature and semilinear PDEs in geometry and topology  2008.5 

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  • Perelman's invariant and the Yamabe invariant

    TUS-NPU Bilatral Seminar 2008 (Proceedings)  2008.3 

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  • On the Yamabe invariant of M x S^1

    The 3rd Geometry Friendship of Japan and Chaina  2008.1 

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  • 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 

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  • Yamabe constants of inifinite coverings and a positive mass theorem

    Variational Problems in Geometry  2007.9 

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  • Yamabe constants of inifinite coverings and a positive mass theorem

    International Coference in Geometry and Analysis, Nanjing 2007  2007.8 

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  • 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 

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  • 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 

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  • 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 

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  • The Yamabe constants of infinite coverings and a positive mass theorem

    Geometry of Singularities  2007.1 

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  • 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 

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  • 山辺不変量について

    筑波大学数学系月例談話会  2006.9 

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  • 山辺不変量と mass 不変量

    東京幾何セミナー  2006.6 

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  • The mass of a compact positive conformal manifold

    リーマン幾何と幾何解析  2006.2 

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  • 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 

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  • 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 

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  • 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 

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  • 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 

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  • 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 

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  • 正質量定理

    ENCOUNTER with MATHEMATICS --数学との遭遇-- 第32回  2005.1 

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  • 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 

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  • 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 

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  • 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 

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  • Yamabe invariants of 3-manifolds

    大域解析学とその周辺  2004.1 

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  • Yamabe invariants of 3-manifolds

    東工大微分幾何国際研究集会  2003.12 

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  • ワイル汎関数と山辺不変量

    筑波大幾何セミナー  2003.9 

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  • 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 

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  • Yamabe metrics on cylindrical manifolds

    The second International Symposium on Differential Geometry  2003.7 

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  • Decomposition along hypersurfaces and the Yamabe invariant of cylindrical manifolds

    Stanford大学幾何セミナー  2003.3 

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  • ワイル汎関数と山辺不変量

    東北大幾何セミナー  2003.1 

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  • 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 

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  • Yamabe metrics on cylindrical manifolds

    Pacific Northwest Geometry/University of Oregon  2001.10 

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Awards

  • 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

Research Projects

  • 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

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    Grant amount: \16250000 ( Direct Cost: \12500000 、 Indirect Cost: \3750000 )

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    Grant amount: \4000000 ( Direct Cost: \3400000 、 Indirect Cost: \600000 )

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  • 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

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    Grant amount: \24570000 ( Direct Cost: \18900000 、 Indirect Cost: \5670000 )

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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.

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  • 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

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    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。

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  • 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

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    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。

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  • 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

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    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。

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  • 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

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    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。

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  • 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

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    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。

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Committee Memberships

  • 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  

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