Value distribution theory of bounded domains

2014 Fiscal Year Final Research Report

• Project/Area Number: 23654021
• 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, 理学(系)研究科(研究院), 教授 (70108182)
• Research Collaborator: KOBAYASHI Ryoichi 名古屋大学, 大学院多元数理科学研究科, 教授 (20162034)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 代数的極小曲面 / 全曲率有限 / ガウス写像 / 除外値 / 全分岐値数 / ネバンリンナ理論 / コーンフォッセンの不等式 / 対数微分の補題

Outline of Final Research Achievements

Through a challenge to estimate the number of exceptional values as well as the total ramified value numbers of the Gauss map of algebraic minimal surfaces, we obtain the following:
1.We found an invariant R given by the ratio of the degree of the Gauss map and a topological quantity, and taking the period condition into account, we have an estimate of the total ramified value number, although the maximal number of the exceptional values is not yet obtained. 2. Lifting all the data to the universal covering (disk) of the surface, we found a candidate of the key number κ, which is somehow corresponds to R above. 3. Using the special properties of minimal surfaces, we consider 1-jet space of the Gauss map to which we apply a new Nevanlinna theory. On the jet space, we consider the proximity function, the counting function and try to use the lemma on logarithmic differential to obtain the final defect relation.

Free Research Field

微分幾何学