Capacities on Levi-flat real hypersurfaces

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K13422
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Shizuoka University
• Principal Investigator: Adachi Masanori 静岡大学, 理学部, 講師 (30708392)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: レビ平坦曲面 / 複素解析幾何 / 葉層構造論 / 多変数関数論 / 力学系理論 / CR幾何学 / 群作用 / 剛性理論

Outline of Final Research Achievements

We investigated Levi-flat real hypersurfaces, important geometric objects in complex analysis in several variables and dynamical systems on holomorphic foliations. Our goal was to contribute to solving the generalized Levi problem and a conjecture by Cerveau. We focused on analyzing the typical examples of Levi-flats, particularly those found within flat ruled surfaces and Inoue-Hirzebruch surfaces. The highlight of our research was successfully confirming a conjecture by Brunella, achieved through collaboration with Judith Brinkschulte.

Free Research Field

複素解析幾何学

Academic Significance and Societal Importance of the Research Achievements

2次元複素射影空間に関するCerveau予想へのアプローチは道半ばとなったが、3次元以上のコンパクト複素多様体における余次元1正則葉層の極小集合の構造論に関して、Brunella予想の解決という2008年以来の大きな進展を得た。当初計画に沿う形でLevi平坦境界の領域の典型例における具体的な解析結果が複数得られた他、Diederich-Fornaess指数のCR幾何学的定式化、カスプ付き双曲曲面に関する松元型剛性定理の新証明など、当初計画で予期しなかった複数の成果も得られた。