Ricci solitons, Yamabe solitons and a generalization of minimal submanifolds

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14534
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Chiba University (2021-2022); Shimane University (2019-2020)
• Principal Investigator: Maeta Shun 千葉大学, 教育学部, 准教授 (00709644)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 山辺ソリトン / コンフォーマルソリトン / リッチ曲率 / ヘッセ多様体 / ヘッセフロー / ヘッセソリトン / 部分多様体 / 双対空間

Outline of Final Research Achievements

I showed the following:
1.Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. 2. I classified 3-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. 3. Any conformal soliton on a hypersurface in a Euclidean space arisen from the position vector field is contained in a hyperplane, a conic hypersurface or a hypersphere. 4. I defined a Hesse soliton, that is, a self-similar solution to the Hesse flow on Hessian manifolds and showed that any compact proper Hesse soliton is expanding and any non-trivial compact gradient Hesse soliton is proper. Furthermore, I showed that the dual space of a Hesse-Einstein manifold can be understood as a Hesse soliton.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

幾何学的フローはポアンカレ予想を含むサーストンの幾何化予想解決に用いられた非常に強力な手法であり，その自己相似解は重要な役割を担う．本研究は幾何学的フローの自己相似解を研究し，いくつかの分類定理を与えたことに意義がある．また，情報幾何で用いられるヘッセ多様体上の幾何学的フローに対して，その自己相似解といくつかの分類を与えたことに意義がある．