Shimpei Kobayashi

CHARACTERIZATIONS OF BIANCHI–BÄCKLUND TRANSFORMATIONS OF CONSTANT MEAN CURVATURE SURFACES

We show that Bianchi–Backlund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.

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⁄ 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.

A Construction Method for Discrete Constant Negative Gaussian Curvature Surfaces

This article is an application of the author’s paper (Kobayashi, Nonlinear d’Alembert formula for discrete pseudospherical surfaces, 2015, [9]) 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 in Lemma 3.1. As an application, we draw figures of discrete constant negative Gaussian curvature surfaces given by this method (Figs. 1 and 2).

On the Bernstein Problem in the Three-dimensional Heisenberg Group

In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group Nil3 by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.