Surfaces with singularities in space-times and Weierstrass-type representation formulas

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400066
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Yamada Kotaro 東京工業大学, 理学院, 教授 (10221657)
• Co-Investigator(Renkei-kenkyūsha): Masaaki Umehara 東京工業大学, 情報理工学院, 教授 (90193945)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: ローレンツ多様体 / ワイエルストラス表現公式 / 特異点 / 極大曲面 / 型変化

Outline of Final Research Achievements

A class of maximal surfaces in Lorentz-Minkowski 3-space, named "Kobayashi surfaces" is introduced. A surface in this class can be extended to a zero-mean curvature surfaces which changes causal types from space-like to time-like. Existence of infinitely many surfaces among this class which are graphs of functions over the space-like plane. On the other hand, the first example of a zero-mean curvature which contains a light-like line and changes causal types across the line is obtained. Such a property, called the light-like line theorem, are generalized for wider class of surfaces.
For a surface in Lorentzian 3-manifold which changes its causal type from space-like to time-like at a light-like point, it is shown that the mean curvature function converges to zero at the light-like point. In particuler, it is shown that a non-zero constant mean curvature surface cannot change its causal type.

Free Research Field

微分幾何学