Showing papers by "Jun-ichi Inoguchi published in 2011"

Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

Abstract: We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $\tau$ function are presented. Backlund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.

Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields

Abstract: Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen’s conjecture or Caddeo, Montaldo and Piu’s one on biharmonic maps into a space of non positive curvature are given. Gauge field analogue is shown, indeed, the isolation phenomena of bi-Yang-Mills fields are obtained.

Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves

Abstract: We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati–Konno–Ichikawa elastic beam equation, the complex Dym equation and the short pulse equation. They are related to the modified KdV or the sine–Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler–Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

Minimal translation surfaces in the Heisenberg group Nil3

Abstract: A translation surface in the Heisenberg group $\mathrm{Nil}_3$ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group $\mathrm{Nil}_3$.

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

Abstract: We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

Parabolic Geodesics in Sasakian $3$-Manifolds

Abstract: We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms Department of Mathematics, Chonnam National University, Gwangju, 500–757, Korea e-mail: jtcho@chonnamackr Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 990-8560, Japan e-mail: inoguchi@scikjyamagata-uacjp Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea e-mail: jieunlee12@gmailcom Received by the editors August 18, 2008 Published electronically March 10, 2011 The third author was partially supported by the National Research Foundation of the Korean Government (NRF-2009-351-C00008) AMS subject classification: 58E20

Curvatures and symmetries of tangent sphere bundles

Abstract: This paper has two purposes. (1) Holomorphic sectional curvature and ?-sectional curvature of tangent sphere bundles are investigated. In particular, tangent sphere bundles of constant holomorphic sectional curvature or of constant ?-sectional curvature are classified. (2) Hypersurface geometry of tangent sphere bundles is developed. Tangent sphere bundles with pseudo-parallel shape operator or ?-parallel shape operator are classified

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

Abstract: In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H 1$. 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 $H eq 0$, 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 Weierstra{\ss} 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 Weierstra{\ss} type representation presented in this paper enables us to construct simultaneously minimal surfaces (H=0) and non-minimal constant mean curvature surfaces ($0