3次元ユークリッド空間内のガウス曲率負一定曲面(以下,K曲面と略記する)は,適切な座標系のもとでサインゴルドン方程式と呼ばれる可積分方程式を持ち,可積分系と微分幾何学の両方の観点から研究されている.さらに,あるK曲面から新たなK曲面を作るベックルンド変換の存在も,K曲面の持つ重要な性質であり,その計算手法の一つとして,Lax対に対するゲージ変換を用いる方法が知られている.一方,Bobenko,Pinkall(1996)およびHoffmann,Sageman-Furnas(2016)はそれぞれ異なる座標系に基づき,K曲面の離散化とLax対の関係を明らかにした.本講演では,Lax対に対するゲージ変換を用いたベックルンド変換の計算手法について,連続と離散の両方の立場から比較しつつ紹介する.本講演の内容の一部はThomas Raujouan氏(University of Tours)とWayne Rossman氏(神戸大学)との共同研究に基づく.
小林 真平(北海道大学) 6/23(火) 10:45-11:45
Generalized discrete isothermic CMC surfaces and their Weierstrass-type representation
In this talk, we present joint work with Tim Hoffmann and Zi Ye on a new framework for discrete isothermic surfaces (2022, Geometriae Dedicata). First, we propose a generalized definition of these surfaces that covers a broader class than previous models. This extension includes associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC) surfaces, effectively capturing the behavior of their smooth counterparts. Second, we demonstrate that discrete isothermic CMC surfaces can be constructed from discrete holomorphic data--specifically, solutions to the additive rational Toda system--via a discrete generalized Weierstrass-type representation.
In this talk, we discuss minimal timelike surfaces in the four-dimensional pseudo-Euclidean space of index two from the viewpoint of para-quaternionic holomorphic geometry.
Minimal timelike surfaces admit parametrizations in terms of null curves.
We explain how these parametrizations can be naturally interpreted in terms of para-quaternions, which provide a pseudo-Riemannian counterpart of quaternionic holomorphic geometry.
This viewpoint clarifies the geometric structure underlying the parametrizations.