Construction of surfaces via complexifications of loop groups

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740042
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hokkaido University (2013); Hirosaki University (2011-2012)
• Principal Investigator: KOBAYASHI Shimpei 北海道大学, 理学(系)研究科(研究院), 准教授 (40408654)
• Project Period (FY): 2011 – 2014
• Keywords: 停留曲面 / ループ群 / 可積分系

Outline of Final Research Achievements

When the structure equations (nonlinear partial differential equations) of a surface is an integrable system, the surface is called "integrable surface". In the research, we gave constructions and characterizations of integrable surfaces. In particular, using loop group structures of integrable surfaces, we gave a construction of minimal surfaces in the three-dimensional Heisenberg group, constant Gaussian curvature surfaces in the three-sphere and a characterization of Demoulin surfaces in the three-dimensional real projective space. Moreover, we gave a new method obtaining the discrete mKdV equation using a loop group action.

Free Research Field

幾何学