Holomorphic representations for discrete surfaces, and analysis on their continuous limit

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14314
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: The University of Tokushima (2022-2023); Kyushu University (2020-2021)
• Principal Investigator: YASUMOTO Masashi 徳島大学, 大学院社会産業理工学研究部(理工学域), 講師 (70770543)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 離散微分幾何 / 可積分系 / 特異点

Outline of Final Research Achievements

It is known that beautiful shapes, such as soap films and soap bubbles, are obtained by certain holomorphic representation formula. In this project, we derived new holomorphic representations representation formulae for discrete surfaces and achieved results in analyzing their continuous limits and asymptotic behaviora. In particular, we were able to uniformly derive holomorphic representation formulae for discrete surfaces based on an integrable systems approach (in collaboration with Mason Pember and Denis Polly), and obtained new insights into the behavior and continuous limit of singularities appearing in discrete zero mean curvature surfaces in three-dimensional Minkowski space.

Free Research Field

離散微分幾何

Academic Significance and Societal Importance of the Research Achievements

コンピュータサイエンスの発展に伴い，従来の微分幾何を，離散的な土台の下で理論を再構築することが強く求められている．特定の曲率条件のもとでの離散曲面・半離散曲面は，様々な数学研究が交差する重要な研究対象であり，従来は3次元ユークリッド空間内の離散化された曲面を中心に研究されてきた．本研究課題では，必ずしも3次元ユークリッド空間とは限らない，より一般の空間内の離散曲面，半離散曲面の数学研究を整備し，種々の離散化された曲面の構成法を新たに導出した．さらに，その無限遠方の振る舞いや特異性を解析する基礎研究の第一歩を新たに発見し，さらに連続極限への収束性についても解析した．