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  2. Katsuhiro Moriya

Katsuhiro Moriya

  (守屋 克洋)

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

 11

Research Areas

 1

Major Research History

 10
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Committee Memberships

 4

Papers

 30
  • Katsuhiro Moriya
    Annals of Global Analysis and Geometry, 61(1) 21-36, Feb, 2022  Peer-reviewedLead authorLast authorCorresponding author
    For 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.
  • K. Leschke, K. Moriya
    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.
  • Katsuhiro, Moriya
    RIMS Kokyuroku, 2152 16-19, Apr, 2020  InvitedLead authorLast authorCorresponding author
  • K. Leschke, K. Moriya
    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.
  • Katsuhiro Moriya
    Springer Proceedings in Mathematics and Statistics, 203 59-68, Sep, 2017  Peer-reviewedLead authorLast authorCorresponding author
    © Springer Nature Singapore Pte Ltd. 2017. A super-conformal map is a conformal map from a two-dimensional Riemannian manifold to the Euclidean four-space such that the ellipse of curvature is a circle. Quaternionic holomorphic geometry connects super-conformal maps with holomorphic maps. We report the Schwarz lemma for super-conformal maps and related results.
  • Kazuyuki Hasegawa, Katsuhiro Moriya
    Advances in Applied Clifford Algebras, 27(2) 1243-1262, Jun 1, 2017  Peer-reviewedLead authorLast authorCorresponding author
    © 2016, Springer International Publishing. A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point.
  • K. Leschke, K. Moriya
    Complex Manifolds, 3(1) 282-300, Jan, 2016  Peer-reviewedLead authorLast authorCorresponding author
    © 2016 K. Leschke and K. Moriya, published by De Gruyter Open. In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez-Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well-known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface.
  • Katsuhiro Moriya
    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.
  • Katsuhiro Moriya
    Geometry, 2013 1-9, 2013  Peer-reviewedLead authorLast authorCorresponding author
    The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäcklund transform. For a given isothermic harmonic inverse mean curvature surface, its classical Darboux transform is a harmonic inverse mean curvature surface. Then a transform of a solution to the Painlevé III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution.
  • Katsuhiro Moriya
    RIMS Kôkyûroku, 1817(1817) 6-10, Nov, 2012  Lead authorLast authorCorresponding author
  • Katsuhiro Moriya
    RIMS Kôkyûroku, 1817(1817) 1-5, Nov, 2012  Lead authorLast authorCorresponding author
  • Sanae Kurosu, Katsuhiro Moriya
    Differential Geometry and its Application, 30(3) 227-232, Jun, 2012  Peer-reviewedLead authorLast authorCorresponding author
    A 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.
  • Katsuhiro Moriya
    Advances in Applied Clifford Algebras, 22(2) 433-448, Jun, 2012  Peer-reviewedLead authorLast authorCorresponding author
    A condition for a closed one-form to be exact, the one-form having values in Euclidean space, on a compact surface without boundary, is given in the case where the surface has suitable differentiable automorphisms. Tori and hyperelliptic curves, with holomorphic automorphisms, are in this case. A local representation formula for surfaces in Euclidean space is then globalized. A condition for a local surface of constant mean curvature to be global, can be written using a harmonic Gauss map. © 2011 Springer Basel AG.
  • Katsuhiro Moriya
    Bulletin of the London Mathematical Society, 41(2) 327-331, Apr, 2009  Peer-reviewedLead authorLast authorCorresponding author
    A 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.
  • Katsuhiro Moriya
    Proceedings of the 16th OCU International Academic Symposium 2008, OCAMI Studies, 3 197--201, Apr, 2009  Peer-reviewedLead authorLast authorCorresponding author
    A surface is represented as a quotient of two quaternionic holomorphic sections. Utilizing these quotients, we explain a correspondence between super- conformal surfaces and complex holomorphic null curves.
  • Katsuhiro Moriya
    RIMS Kôkyûroku, 1623(1623) 30-34, Jan, 2009  Lead authorLast authorCorresponding author
  • Katsuhiro Moriya
    Annals of Global Analysis and Geometry, 34(1) 1-20, Aug, 2008  Peer-reviewedLead authorLast authorCorresponding author
    A 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.
  • Katsuhiro Moriya
    RIMS Kôkyûroku, 1577(1577) 36-44, Jan, 2008  Lead authorLast authorCorresponding author
  • Katsuhiro Moriya
    RIMS Kôkyûroku, 1527(1527) 128-134, Jan, 2007  Lead authorLast authorCorresponding author
  • Katsuhiro Moriya
    Tsukuba Journal of Mathematics, 30(1) 131-135, Jun, 2006  Peer-reviewedLead authorLast authorCorresponding author
    We will explicitly give the defining equation of the moduli space of symmetric minimal tori with one end.
  • Katsuhiro Moriya
    Proceedings of the American Mathematical Society, 131(1) 303-307, Jan, 2003  Peer-reviewedLead authorLast authorCorresponding author
    We 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.
  • Katsuhiro Moriya
    数理解析研究所講究録, 1292(1292) 12-19, Oct, 2002  Lead authorLast authorCorresponding author
  • Katsuhiro Moriya
    Geom. Integrability & Quantization, Proceedings of the Third International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov and Gregory L. Naber, eds. (Sofia: Coral Press Scientific Publishing, 2002),, 360-368, 2002  Lead authorLast authorCorresponding author
    In this paper, we will report a recent study about moduli spaces of branched and complete minimal surfaces in Euclidean space of genus one with two ends and total curvature −4π.
  • Katsuhiro Moriya
    Osaka Journal of Mathematics, 38(2) 271-285, Jun, 2001  Peer-reviewedLead authorLast authorCorresponding author
  • Katsuhiro Moriya
    Josai mathematical monographs, 3 149-154, Mar, 2001  Peer-reviewedLead authorLast authorCorresponding author
    We will report our recent result on existence of a complex one-parameterfamily of complete minimal surfaces of genus one with one end and finitetotal curvature. The family connects a minimal surface with total curvature -12πand that with total curvature less than -12π.
  • Katsuhiro Moriya
    Journal of mathematical sciences, the University of Tokyo, 7(2) 311-323, 2000  Peer-reviewed
    In this paper, we will prove that a certain class of branched multi-valued minimal surfaces invariant under a translation or a screw motion becomes a real analytic variety via their Weierstrass data. We also prove that the class contains complex analytic variety and give a lower bound of its dimension.
  • 守屋, 克洋
    RIMS Kôkyûroku, 1113(1113) 35-43, Oct, 1999  Lead authorLast authorCorresponding author
  • Katsuhiro, Moriya
    Differential geometry and applications (Brno, 1998), 207--213, Jan, 1999  Peer-reviewed
  • Katsuhiro Moriya
    RIMS Kôkyûroku, 1044(1044) 63-75, Apr, 1998  Lead authorLast authorCorresponding author
  • Katsuhiro Moriya
    Tokyo Journal of Mathematics, 21(1) 121-134, 1998  Peer-reviewedLead authorLast authorCorresponding author
    In 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

 5

Major Presentations

 107
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Major Teaching Experience

 69
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Professional Memberships

 2

Major Research Projects

 15
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Other

 5
  • Jan, 2018 - Jan, 2018
  • Mar, 2017 - Mar, 2017
    A 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
  • Sep, 2016 - Sep, 2016
    A 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
  • Apr, 2013 - Apr, 2013
    四元数複素微分幾何学リサーチグループ代表 https://sites.google.com/site/qcdgrg/
  • Apr, 2013 - Apr, 2013
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