Scala-flat complete Kaehler metrics and K-stability at infinity

2017 Fiscal Year Final Research Report

• Project/Area Number: 26610015
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: Kobayashi Ryoichi 名古屋大学, 多元数理科学研究科, 教授 (20162034)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 代数的極小翼面 / 周期条件 / 自由フックス群 / 放物型局所化原理 / ネヴァンリンナ理論 / 対数微分補題 / ガウス写像 / 力学系

Outline of Final Research Achievements

By replacing an albebraic minimal surface by the triple consisting of the disc (the universal cover), the action of free Fuchsian group (fundamental group) and the period condition, we formulate a theory for all algebraic minimal surfaces. By applying the free Fuchsian group to a fixed fundamental domain, we get a dynamical system on the circle consisting of increasing number of vertices of fundamental domains whose evolution is governed by increasing word length. The properties of this dynamical system was formulated as the parabolic localization principle. I propose a pair of a meromorphic function exp(H) and a potentially infinite degree divisor D on the Riemann sphere, (exp(H),D), which encodes the period condition. The parabolic localization principle implies that the Gauss map is in the maximal approximation state relative to D. So is exp(H). It follows that the LLD (lemma on log derivative) applied to the pair (exp(H),D) decodes the period condition.

Free Research Field

複素幾何学