研究者業績
Profile Information
- Affiliation
- Professor, Graduate School of Science, University of Hyogo
- Degree
- Doctor of Science(Sep, 1999, Tokyo Metropolitan University)
- ORCID ID
https://orcid.org/0000-0001-8061-5264- J-GLOBAL ID
- 200901060625012121
- Researcher ID
- B-6244-2014
- researchmap Member ID
- 1000295662
Research Interests
11Research Areas
1Major Research History
10-
Apr, 2021 - Present
Committee Memberships
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Aug, 2013 - Jul, 2015
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2006 - 2006
Major Papers
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Annals of Global Analysis and Geometry, 61(1) 21-36, Feb, 2022 Peer-reviewedLead authorLast authorCorresponding authorFor a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson’s polar variety and bipolar surface.
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Manuscripta Mathematica, 162(3-4) 537-558, Jul 1, 2020 Peer-reviewedLead authorLast authorCorresponding author© 2019, The Author(s). The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated C∗-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called μ-Darboux transforms. We show that a μ-Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a μ-Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in CP3 which is canonically associated to a minimal surface fp,q in the right-associated family of f. Here we use an extension of the notion of the associated family fp,q of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at μ= 1. Moreover, the family of Willmore surfaces μ-Darboux transforms, μ∈ C∗, extends to a CP1 family of Willmore surfaces fμ: M→ S4 where μ∈ CP1.
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Mathematische Zeitschrift, 291(3-4) 1015-1058, Apr, 2019 Peer-reviewedLead authorLast authorCorresponding author© 2018, The Author(s). The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.
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Israel Journal of Mathematics, 207(1) 331-359, Apr, 2015 Peer-reviewedLead authorLast authorCorresponding author© 2015, Hebrew University of Jerusalem. A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a meromorphic function. Analogs of the Liouville theorem, the Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem, the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and branch points of a super-conformal map are obtained.
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Differential Geometry and its Application, 30(3) 227-232, Jun, 2012 Peer-reviewedLead authorLast authorCorresponding authorA tt *-bundle is constructed by a harmonic map from a Riemann surface into an n-dimensional sphere. This tt *-bundle is a high-dimensional analogue of a quaternionic line bundle with a Willmore connection. For the construction, a flat connection is decomposed into four parts by a fiberwise complex structure. © 2012 Elsevier B.V.
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Bulletin of the London Mathematical Society, 41(2) 327-331, Apr, 2009 Peer-reviewedLead authorLast authorCorresponding authorA correspondence from a null complex holomorphic curve in four-dimensional complex Euclidean space to a super-conformal surface in four-dimensional Euclidean space is defined by the quaternionic theory of surfaces. As an application, a transformation of super-conformal surfaces is defined. © 2009 London Mathematical Society.
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Annals of Global Analysis and Geometry, 34(1) 1-20, Aug, 2008 Peer-reviewedLead authorLast authorCorresponding authorA quotient of two linearly independent quaternionic holomorphic sections of a quaternionic holomorphic line bundle over a Riemann surface is a conformal branched immersion from a Riemann surface to four-dimensional Euclidean space. On the assumption that a quaternionic holomorphic line bundle is associated with a Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane, we shall classify the denominators of Lagrangian-branched immersion from a Riemann surface to complex Euclidean plane. © 2008 Springer Science+Business Media B.V.
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Tsukuba Journal of Mathematics, 30(1) 131-135, Jun, 2006 Peer-reviewedLead authorLast authorCorresponding authorWe will explicitly give the defining equation of the moduli space of symmetric minimal tori with one end.
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Proceedings of the American Mathematical Society, 131(1) 303-307, Jan, 2003 Peer-reviewedLead authorLast authorCorresponding authorWe will show that any punctured Riemann surface can be conformally immersed into a Euclidean 3-space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.
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Tokyo Journal of Mathematics, 21(1) 121-134, 1998 Peer-reviewedLead authorLast authorCorresponding authorIn this paper, we show that the moduli space of the Weierstrass data for algebraic minimal surfaces in Euclidean 4-space with fixed topological type, orders of branched points and ends, and total curvature, has the structure of a real analytic variety. We provide the lower bounds of its dimension. We also show that the moduli space of the Weierstrass data for stable algebraic minimal surfaces in Euclidean 4-space has the structure of a complex analytic variety. © 1998 by the University of Notre Dame. All rights reserved.
Books and Other Publications
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Institute of Mathematics, University of Tsukuba, 2007
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Institute of Mathematics, University of Tsukuba, 2006
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Institute of Mathematics, University of Tsukuba, 2003
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Translations of Mathematical Monographs American Mathematical Society, Jan, 2003
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Institute of Mathematics, University of Tsukuba, 2002
Major Presentations
107-
The 3rd Shot of The 13th MSJ-SI "Differential Geometry and Integrable Systems", Mar 4, 2023, Yoshihiro Ohnita (OCAMI Osaka Metropolitan University), Franz Pedit (UMass Amherst), Martin Guest (Waseda University) Toru Kajigaya (Tokyo University of Science) Wayne Rossman (Kobe University) Masa-Hiko Saito (Kobe Gakuin University) Takashi Sakai (chair, Tokyo Metropolitan University) Masashi Yasumoto (secretary, Tokushima University) Invited
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Aspects of submanifolds and other related fields, Jun 25, 2018, Satoshi Kawakubo Invited
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The 20th International Workshop on Hermitian Symmetric Spaces and Submanifolds, Jul 29, 2016, Young Jin Suh, Yoshihiro Ohnita, Jiazu Zhou, Byung Hak Kim InvitedA super-conformal map from a Riemann surface to the Euclidean four-space is a surface with circular ellipse of curvature with respect to the induced metric. This map has properties similar to the holomorphic function on a Riemann surface. In this talk, the Schwarz lemma for super-conformal maps is explained.
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MSJ Spring Meeting 2016 at University of Tsukuba, Mar 16, 2016, The Mathematical Society of JapanWe factorize a super-conformal map. This factorization connects a super-conformal map with a holomorphic map. Then we obtain the Schwarz–Pick theorem for super-conformal maps. Then we define a distance on the image of a super-conformal map.
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MSJ Spring Meeting 2016 at University of Tsukuba, Mar 16, 2016, The Mathematical Society of JapanIn this talk, we take up two classes of conformal maps and apply the canonical factorization. One is constrained Willmore surfaces and the other is minimal surfaces. A factor of a canonical factorization for a conformal map provides a canonical lift of a conformal map. We characterize constrained Willmore surfaces by canonical lifts. A factor of a canonical factorization for a conformal map provides the area element of a conformal map. We give an upper bound of the area of a minimal surface around a branch point.
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MSJ Autumn Meeting 2010, at Nagoya University, Sep 24, 2010, The Mathematical Society of Japan
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Riemann Surfaces, Harmonic Maps and Visualization, Dec 19, 2008 Invited
Major Teaching Experience
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Apr, 2022 - Mar, 2023graduation research (University of Hyogo)
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Oct, 2022 - Feb, 2023Probability and Statistics (University of Hyogo)
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Apr, 2022 - Aug, 2022フォトンサイエンス特論 (兵庫県立大学)
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Apr, 2022 - Aug, 2022Algebra I (University of Hyogo)
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Oct, 2021 - Feb, 2022analysis II (University of Hyogo)
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Apr, 2021 - Aug, 2021Science English (University of Hyogo)
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Apr, 2021 - Aug, 2021Exercises in Mathematics I (University of Hyogo)
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Apr, 2021 - Aug, 2021geometric structure (University of Hyogo)
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Apr, 2021 - Aug, 2021algebraic structure (University of Hyogo)
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Oct, 2020 - Jan, 2021Linear Algebra II (University of Hyogo)
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Oct, 2020 - Jan, 2021Analysis II (University of Hyogo)
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Apr, 2019 - Mar, 2020卒業研究 (兵庫県立大学)
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Oct, 2018 - Feb, 2019Introduction to manifolds (University of Tsukuba)
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Oct, 2018 - Jan, 2019Exercise in manifolds (University of Tsukuba)
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Apr, 2018 - Jul, 2018Problems for Students in Introduction to Differential Equations (University of Tsukuba)
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Apr, 2018 - Jul, 2018Mathematics in Foreign Language I (University of Tsukuba)
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線形代数 I 演習 (筑波大学)
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微積分 II 演習 (筑波大学)
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ベクトル解析と幾何演習 (筑波大学)
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線形代数続論演習 (筑波大学)
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数学外書輪講II (筑波大学)
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数学外書輪講I (筑波大学)
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フレッシュマン・セミナー (筑波大学)
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数学類特別セミナーI (筑波大学)
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微積分II演習 (筑波大学)
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クラスセミナー (筑波大学)
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数学類特別セミナーII (筑波大学)
Professional Memberships
2-
Jan, 2012 - Present
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Jun, 1999 - Present
Major Research Projects
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Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science, Apr, 2022 - Mar, 2027
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Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, Apr, 2017 - Mar, 2022
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Grant-in-Aid for Scientific Research (C), Japan Society of for the Promotion of Science, Apr, 2017 - Mar, 2021
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International Networks, The Leverhulme Trust, Sep, 2016 - Aug, 2018
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Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, Apr, 2013 - Mar, 2017
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Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, Apr, 2010 - Mar, 2012
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Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B), Japan Society for the Promotion of Science, Apr, 2007 - Mar, 2008
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Grant-in-Aid for Young Scientists (B), Japan Society of for the Promotion of Science, Apr, 2004 - Mar, 2006
Other
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Mar, 2017 - Mar, 2017A member of the organising committee of m:iv spring 2017 workshop at University College Cork in Ireland. http://www2.le.ac.uk/projects/miv/workshop-programme/spring-2017-workshop
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Sep, 2016 - Sep, 2016A network partner of the m:iv minimal surfaces: integrable systems and visualisation. An international research group funded by The Leverhulme Trust. Led by Dr Katrin Leschke at the University of Leicester, Department of Mathematics; m:iv brings together researchers at five international institutions to work on the study of minimal surfaces: combining the expertise of the network partners in the areas of visualisation, minimal surfaces and integrable systems will allow new approaches in this research area. The network will run a series of seminars, ranging from introductory presentations to detailed talks on specialised results. The seminar series will develop the necessary foundations for the research whilst computer experiments are undertaken. Extended research visits each year will take place between network partners. In addition to the seminar series and research visits, the network will run a programme of workshops, hosted in turn by each network partner highlighting their area of research, with the final workshop taking place at Leicester, where the various strands will be linked together. http://www2.le.ac.uk/projects/miv The University has agreed to host members of the