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Shimpei Kobayashi

Publications and source records attributed to Shimpei Kobayashi.

34 records · Page 2Linked to original sources
arXiv2020

Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$, as well as definite and indefinite affine spheres in $\mathbb R^3$. In this paper we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space $\mathbb{CH}^2_1$. We show that this class of surfaces corresponds to the fifth real form. Moreover, for each timelike Lagrangian surface in $\mathbb {CH}^2_1$ we define natural Gauss maps into certain homogeneous spaces and prove a Ruh-Vilms type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.

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arXiv2020

Discrete Constant Mean Curvature Surfaces on General Graphs

The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.

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arXiv2020

Representation formula for discrete indefinite affine spheres

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.

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arXiv2019

General-affine invariants of plane curves and space curves

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bf R})\ltimes {\bf R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bf R})\ltimes {\bf R}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.

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arXiv2019

Survey on real forms of the complex $A_2^{(2)}$-Toda equation and surface theory

The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for $k$-symmetric spaces over reductive Lie groups. In this survey we will show that to each of the five different types of real forms for a loop group of $A_2^{(2)}$ there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in $\mathbb {CP}^2$, minimal Lagrangian surfaces in $\mathbb {CH}^2$, timelike minimal Lagrangian surfaces in $\mathbb {CH}^2_1$, proper definite affine spheres in $\mathbb R^3$ and proper indefinite affine spheres in $\mathbb R^3$, respectively.

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arXiv2017

Nonlinear d'Alembert formula for discrete pseudospherical surfaces

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.

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arXiv2016

A construction method for discrete constant negative Gaussian curvature surfaces

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.

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arXiv2015

Constant mean curvature surfaces in hyperbolic 3-space via loop groups

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H \neq 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0 \leq H < 1$ 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 $\mathbb{H}^3$ with particular emphasis on the case of mean curvature $0\leq H < 1$. 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 ($0<H<1$).

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arXiv2015

A loop group method for minimal surfaces in the three-dimensional Heisenberg group

We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle $\D \times \GL$ over a simply connected domain $\mathbb{D}$ 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.

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arXiv2013

A loop group method for Demoulin surfaces in the 3-dimensional real projective space

For a surface in the 3-dimensional real projective space, we define a Gauss map, which is a quadric in $\mathbb R^{4}$ and called the first-order Gauss map. It will be shown that the surface is a Demoulin surface if and only if the first-order Gauss map is conformal, and the surface is a projective minimal coincidence surface or a Demoulin surface if and only if the first-order Gauss map is harmonic. Moreover for a Demoulin surface, it will be shown that the first-order Gauss map can be obtained by the natural projection of the Lorentz primitive map into a 6-symmetric space. We also characterize Demoulin surfaces via a family of flat connections on the trivial bundle $\D \times \SL$ over a simply connected domain $\mathbb{D}$ in the Euclidean 2-plane.

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arXiv2013

Constant Gaussian curvature surfaces in the 3-sphere via loop groups

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 $K\neq 0$, 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 $0<K<1$.

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arXiv2012

Real forms of complex surfaces of constant mean curvature

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 $Λ\mathfrak{sl}(2, \mathbb C)_σ$, and classify all such surfaces according to the classification of real forms of $Λ\mathfrak{sl}(2, \mathbb C)_σ$. 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 Gauß 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 Weierstraß type representation.

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arXiv2009

Loop Group Methods for Constant Mean Curvature Surfaces

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.

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