Researcher Database

Shimpei Kobayashi
Faculty of Science Mathematics Mathematics
Associate Professor

Researcher Profile and Settings

Affiliation

  • Faculty of Science Mathematics Mathematics

Job Title

  • Associate Professor

J-Global ID

Research Interests

  • 可積分系   平均曲率一定曲面   線形常微分   線型常微分方程式   ループ群   flat surface   量子コホモロジー   spacelike surfaces   トポロジー   微分幾何   Euclidean 3-space   平均曲率   特異点   hyperbolic 3-space   de Sitter 3-space   幾何学   flat surfaces   ガウス曲率   constant mean curvature   可視化   コンピュータ実験   Minkowski 3-space   spacelike surface   曲面論   Surfaces theory   

Research Areas

  • Natural sciences / Geometry

Academic & Professional Experience

  • 2013/09 - Today Hokkaido University
  • 2011/04 - 2013/08 Hirosaki University
  • 2008/04 - 2011/03 Hirosaki University
  • 2005/04 - 2008/03 Tokyo Denki University

Research Activities

Published Papers

  • Shimpei Kobayashi
    JOURNAL OF GEOMETRY AND PHYSICS 119 208 - 223 0393-0440 2017/09 [Refereed][Not invited]
     
    On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature (pseudospherical) surfaces in Euclidean three space. We also compute two examples by this formula in detail. (C) 2017 Elsevier B.V. All rights reserved.
  • Josef F. Dorfraeister, Jim-ichi Inoguchi, Shimpei Kobayashi
    ADVANCES IN MATHEMATICS 298 207 - 253 0001-8708 2016/08 [Refereed][Not invited]
     
    We generalize the UhlenbeckSegal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection. (C) 2016 Elsevier Inc. All rights reserved.
  • Josef F. Dorfmeister, Jun-Ichi Inoguchi, Shimpei Kobayashi
    ASIAN JOURNAL OF MATHEMATICS 20 (3) 409 - 448 1093-6106 2016/07 [Refereed][Not invited]
     
    We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle D x GL(2)C over a simply connected domain D in the complex plane. In particular for minimal surfaces, we give an immersion formula, the so-called Sym-formula, and a generalized Weierstrass type representation via the loop group method. Our generalized Weierstrass type representation produces all simply-connected non-vertical minimal surfaces in the Heisenberg group.
  • Shimpei Kobayashi
    Mathematical Progress in Expressive Image Synthesis III, Extended and Selected Results from the Symposium MEIS2015 21 - 33 2016/04 [Not refereed][Not invited]
     
    This article is an application of the author's paper about a construction
    method for discrete constant negative Gaussian curvature surfaces, the
    nonlinear d'Alembert formula. The heart of this formula is the Birkhoff
    decomposition, and we give a simple algorithm for the Birkhoff decomposition.
    As an application, we draw figures of discrete constant negative Gaussian
    curvature surfaces given by this method.
  • Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi
    CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES 59 (1) 50 - 61 0008-4395 2016/03 [Refereed][Not invited]
     
    In this note we present a simple alternative proof for the Bernstein problem in the threedimensional Heisenberg group Nil(3) by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed AbreschRosenberg differential.
  • Shimpei Kobayashi
    DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 40 57 - 66 0926-2245 2015/06 [Refereed][Not invited]
     
    A natural Gauss map for a surface in the 3-dimensional real projective space P-3 will be defined and called the first-order Gauss map. It will be shown that the first-order Gauss map is conformal if and only if it is a Demoulin surface, which is a special case among projective minimal surfaces. Moreover, it will be shown that the first-order Gauss map is Lorentz harmonic if and only if it is a Demoulin surface or a projective minimal coincidence surface. We also characterize the surfaces via a family of flat connections on the trivial bundle D x SL4R over a simply connected domain D in the Euclidean 2-plane. (C) 2015 Elsevier B.V. All rights reserved.
  • David Brander, Jun-ichi Inoguchi, Shimpei Kobayashi
    PACIFIC JOURNAL OF MATHEMATICS 269 (2) 281 - 303 0030-8730 2014/06 [Not refereed][Not invited]
     
    In this paper we study constant positive Gauss curvature K surfaces in the 3-sphere S-3 with 0 < K < 1, as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in S-3 with Gauss curvature K < 1 is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if K is constant. We give a uniform loop group formulation for all such surfaces with K not equal 0, and use the generalized d'Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with K < 0 and those with 0 < K < 1.
  • Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi
    JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK 686 1 - 36 0075-4102 2014/01 [Refereed][Not invited]
     
    In hyperbolic 3-space H-3 surfaces of constant mean curvature H come in three types, corresponding to the cases 0 <= H < 1, H = 1, H > 1. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space E-3 with H = 0 and H not equal 0, respectively. These surface classes have been investigated intensively in the literature. For the case 0 <= H < 1 there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstrass type representation for surfaces of constant mean curvature in H-3 with particular emphasis on the case of mean curvature 0 <= H < 1. In particular, the generalized Weierstrass type representation presented in this paper enables us to construct simultaneously minimal surfaces (H = 0) and non-minimal constant mean curvature surfaces (0 < H < 1).
  • Discretization of integrable systems via dressing actions.
    KOBAYASHI Shimpei
    RIMS Kôkyûroku Bessatsu B41 161 - 171 2013 [Refereed][Not invited]
  • Shimpei Kobayashi
    Transactions of the American Mathematical Society 363 (2011), no. 4, 1765--1788 2012/03/08 [Not refereed][Not invited]
     
    It is known that complex constant mean curvature ({\sc CMC} for short)
    immersions in $\mathbb C^3$ are natural complexifications of {\sc
    CMC}-immersions in $\mathbb R^3$. In this paper, conversely we consider {\it
    real form surfaces} of a complex {\sc CMC}-immersion, which are defined from
    real forms of the twisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra $\Lambda
    \mathfrak{sl}(2, \mathbb C)_\sigma$, and classify all such surfaces according
    to the classification of real forms of $\Lambda \mathfrak{sl}(2, \mathbb
    C)_\sigma$. There are seven classes of surfaces, which are called {\it
    integrable surfaces}, and all integrable surfaces will be characterized by the
    (Lorentz) harmonicities of their Gau{\ss} maps into the symmetric spaces $S^2$,
    $H^2$, $S^{1,1}$ or the 4-symmetric space $SL(2, \mathbb C)/U(1)$. We also give
    a unification to all integrable surfaces via the generalized Weierstra{\ss}
    type representation.
  • Shimpei Kobayashi
    BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 82 (2) 240 - 253 0004-9727 2010/10 [Refereed][Not invited]
     
    We detail a construction of totally symmetric surfaces of constant mean curvature 0 <= H < 1 in hyperbolic 3-space of sectional curvature -1 via the generalized Weierstrass type representation.
  • Complex surfaces of constant mean curvature fibered by minimal surfaces
    Josef Dorfmeister, Shimpei Kobayashi, Franz Pedit
    HOKKAIDO MATHEMATICAL JOURNAL 39 (1) 1 - 55 0385-4035 2010/02 [Refereed][Not invited]
     
    We define complex constant mean curvature immersions in complex three space using a natural extension of constant mean curvature immersions in Euclidean three space via loop group techniques. We then discuss the fundamental properties of these complex constant mean curvature immersions. In particular, we prove that these immersions are doubly ruled by holomorphic null curves. We present a construction of minimal immersions from constant mean curvature immersions in Euclidean three space via the associated complex constant mean curvature immersions.
  • Shimpei Kobayashi
    ANNALS OF GLOBAL ANALYSIS AND GEOMETRY 36 (4) 375 - 380 0232-704X 2009/12 [Refereed][Not invited]
     
    We give a simple criterion for equivariant harmonic maps into complex projective spaces CP (n) . As an application of the criterion, we give examples of equivariant harmonic cylinders. We also give examples of non-equivariant harmonic cylinders as perturbations of equivariant harmonic cylinders.
  • Shimpei Kobayashi
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 136 (4) 1433 - 1443 0002-9939 2008 [Refereed][Not invited]
     
    A constant mean curvature surface with bubbletons is defined by the loop group action on the set of extended framings for constant mean curvature surfaces by simple factors. Classically such surfaces were obtained by the transformation of tangential line congruences, the so- called Bianchi-Backlund transformations. In this paper, we consider constant mean curvature surfaces with Delaunay ends in three- dimensional space forms R-3, S-3 and H-3 and their surfaces with bubbletons for which the topology is preserved. We show that the ends of such surfaces are again asymptotic to Delaunay surfaces.
  • N. Schmitt, M. Kilian, S.-P. Kobayashi, W. Rossman
    JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 75 563 - 581 0024-6107 2007/06 [Refereed][Not invited]
     
    A theorem on the unitarizability of loop group valued monodromy representations is presented and applied to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply connected 3-dimensional space forms R-3, S-3 and H-3. Additionally, the extended frame for any associated family of Delaunay surfaces is computed.
  • J. Dorfmeister, S. -P. Kobayashi
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 359 (6) 2483 - 2500 0002-9947 2007 [Refereed][Not invited]
     
    We give a coarse classification of constant mean curvature (CMC) immersions of cylinders into R-3 via the loop group method. Particularly for this purpose, we consider double loop groups and a new type of "potentials" which are meromorphic 1-forms on Riemann surfaces.
  • M Kilian, SP Kobayashi, W Rossman, N Schmitt
    JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 72 258 - 272 0024-6107 2005/08 [Refereed][Not invited]
     
    The paper shows the existence of several new families of noncompact constant mean curvature surfaces: (i) singly punctured surfaces of arbitrary genus g >= 1, (ii) doubly punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.
  • S Kobayashi, J Inoguchi
    INTERNATIONAL JOURNAL OF MATHEMATICS 16 (2) 101 - 110 0129-167X 2005/02 [Refereed][Not invited]
     
    We show that Bianchi-Backlund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.
  • Bubbletons in 3-dimensional space forms
    KOBAYASHI Shimpei
    Balkan Journal of Geometry and its Applications 9 (1) 44 - 68 2004 [Refereed][Not invited]

MISC

  • Shoichi Fujimori, Shimpei Kobayashi, Wayne Rossman  Rokko Lectures in Mathematics 17, October 2005  2006/02/25  [Not refereed][Not invited]
     
    This is an elementary introduction to a method for studying harmonic maps
    into symmetric spaces, and in particular for studying constant mean curvature
    (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu. There
    already exist a number of other introductions to this method, but all of them
    require a higher degree of mathematical sophistication from the reader than is
    needed here. The authors' goal was to create an exposition that would be
    readily accessible to a beginning graduate student, and even to a highly
    motivated undergraduate student. Constant mean curvature surfaces in Euclidean
    3-space, and also spherical 3-space and hyperbolic 3-space, are described,
    along with the Lax pair equations that determine their frames. The simplest
    examples, including Delaunay surfaces and Smyth surfaces, are described in
    detail.

Research Grants & Projects

  • 文部科学省:科学研究費補助金(若手研究(B))
    Date (from‐to) : 2011 -2011 
    Author : 小林真平
  • 文部科学省:科学研究費補助金(若手研究(B))
    Date (from‐to) : 2008 -2010 
    Author : 小林真平
     
    1.平均曲率一定曲面の複素化である複素平均曲率一定曲面の一般論についての論文をHokkaido Mathematical Journalに発表し(Josef Dorfmeister氏とFranz Pedit氏との共同研究)、その実形についての論文をTransactions of the American Mathematical Societyに発表予定である。2.複素平均曲率一定曲面の実形として得られる曲面の中で、3次元双曲空間内の極小曲面は他とは大きく性質が異なり、ガウス写像が4-対称空間へのある性質を持つ調和写像になっている。この曲面に対して、Weierstrassの表現公式を確立し、具体例を構成した。それらについてJosef Dorfmeister氏、井ノ口順一氏と共同で論文を纏め現在投稿中である。また位相的に特別な性質をもつ極小曲面を構成し、それについて纏め現在投稿中である。3.複素射影空間内の同変な調和写像の特徴付けを得る事ができた。これらすべての同変調和写像は、次数が1の多項式のループ環に値を持つ微分形式を用いて構成される。この結果をAnnals of Global Analysis and Geometryに発表した。

Educational Activities

Teaching Experience

  • Geometry A
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 多様体, 可微分写像、ベクトル場、微分形式
  • Calculus II
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 原始関数, 積分, 重積分, リ-マン和, 変数変換
  • Exercises on Geometry
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 多様体,ホモロジー


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