A new development of constant mean curvature surfaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400062
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Kenmotsu Katsuei 東北大学, 理学(系)研究科(研究院), 名誉教授 (60004404)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 平行平均曲率ベクトル場 / 複素空間形 / 平均曲率一定曲面

Outline of Final Research Achievements

We proved that a parallel mean curvature surface in the non-flat complex space form of complex two dimension, in general, depends on one harmonic function and five real constants. It is the point for the proof to use the Kaehler angle function as one of the coordinates of the surface. In fact, the first and second fundamental forms of the surface can be written in terms of the Kaehler angle function. Moreover, it is proved that the Kaehler angle function is given as an integral transformation of a harmonic function on the surface. Applying these results to the case that the surface is homeomorphic to a torus, we proved that the immersion must be totally real, and hence we determined those tori explicitly.

Free Research Field

微分幾何学