Geometric value distribution theory

2017 Fiscal Year Final Research Report

• Project/Area Number: 26610011
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Miyaoka Reiko 東北大学, 高度教養教育・学生支援機構, 総長特命教授 (70108182)
• Research Collaborator: Kobayashi Ryoichi
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 極小曲面 / ガウス写像 / 除外値問題 / ネバンリンナ理論 / 普遍被覆 / 基本群作用

Outline of Final Research Achievements

The conjecture that the Gauss map of an algebraic minimal surface M in the Euclidean 3-space omits at most 2 points is a long standing problem. We lift the Gauss map to the universal covering of the surface, which is invariant under the action of the fundamental group of M, and apply the Nevanlinna theory to the lifted g.
1. We obtain the upper bound of κ which estimate the growth order of the characteristic function of g. 2. The ratio of the spherical area of the image of the Gauss map from the universal covering, and the hyperbolic area of the disk is bounded below by certain way. 3. Clarify the geometric meaning of the Lemma on logarithmic derivative. 4.We put the coordinate function as a power of the exponential function, then translate the no real period condition into that the absolute value of the exponential function is invariant. This is a use of the period condition in the most effective way to induce a defect relation.

Free Research Field

微分幾何学