A new approach for the existence problem of the complex structure by applying parabolic flows

2020 Fiscal Year Final Research Report

• Project/Area Number: 19K14543
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kagawa National College of Technology (2020); Kochi National College of Technology (2019)
• Principal Investigator: Kawamura Masaya 香川高等専門学校, 一般教育科(高松キャンパス), 講師 (90805673)
• Project Period (FY): 2019-04-01 – 2021-03-31
• Keywords: 放物型フロー / 複素構造 / 概複素幾何学 / チャーン接続

Outline of Final Research Achievements

In the first year of this research project, we mainly tackled the problem of the regularity of the almost Hermitian curvature flow (AHCF). We investigated some properties of solutions to the flow (AHCF) such as the higher order regularity. We studied the obstacle for the existence of the long time solution, and we then could show the obstacle by using the torsion and the curvature. In the second year, we gave a proof of the uniform equivalence between a solution of (AHCF) and an arbitrary chosen almost Hermitian metric. Based on these results, we have been working on proving that some positivity of the Chern-Ricci curvature can be preserved along (AHCF). By applying the positivity preserved along the flow, we will show the relation between the curvature and the integrability of the almost complex structure.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

概複素構造の積分可能性については, 知られていない部分が多く, 例えば6次元球面に複素構造が入るかどうかという問題は約70年間未解決の問題である. 私の研究は, 放物型フローを用いて, 曲率と概複素構造の積分可能性の関係性を明らかにしようという試みであり, これまでとは異なるアプローチによって, 今まで得られなかった新しい結果が得られる可能性を多く秘めている. 今回の研究により, 概Hermitian曲率フローの持つ性質が幾つか明らかになり, これらの結果を基にして曲率の正性の保存性などの研究が現在も進行中である.