Study of properties of solutions to geometric higher order variational problems

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14341
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Miura Tatsuya 東京工業大学, 理学院, 准教授 (40838744)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 幾何学的変分問題 / 高階問題 / 弾性曲線 / 弾性流 / p-弾性曲線 / 極小曲面 / Topping 予想 / 距離関数

Outline of Final Research Achievements

Among geometric variational problems, we studied problems that include higher-order derivatives such as curvature in their energy, focusing on the properties of the solutions. In particular, we focused on the problem of elastic curves, which is a variational problem of the bending energy that measures how a curve bends, and obtained results such as geometric inequalities, classification theorems for critical points, and analysis of the behavior of gradient flows.
In the process, we also worked extensively on related problems of geometric analysis. We partly solved the Topping conjecture on the relation between the diameter of a surface and its mean curvature, and found an application to the theory of minimal surfaces. We also proved a general structure theorem on the singular set of the distance function.

Free Research Field

数理解析学関連

Academic Significance and Societal Importance of the Research Achievements

弾性曲線の研究は Daniel Bernoulli および Leonhard Euler により 18 世紀に創始されたものであり、高階幾何学的変分問題の最も基本的な例として純粋数学的に重要であるのみならず、弾性棒の形状を中心とした物理現象の解析に直接適用可能であることや、画像処理などの応用分野においても重要な役割を果たすことが知られている。このような古典的問題を含む様々な幾何解析の問題に対し、未解決問題の解決を含む種々の新しい成果が得られたことは、学術的にも社会的にも意義深いものと考えられる。