Classification of biharmonic maps and biharmonic submanifolds, and its applications

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K17542
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Shimane University
• Principal Investigator: Maeta Shun 島根大学, 学術研究院理工学系, 講師 (00709644)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 2重調和写像 / 2重調和部分多様体 / Chen予想 / BMO予想 / 3重調和 / 沈め込み

Outline of Final Research Achievements

I gave many affirmative partial answers to BMO conjecture that is, any complete biharmonic submanifold M in spheres has constant mean curvature" under the following assumptions: 1. the sectional curvature of M is bounded from above and the mean curvature is bounded from below with some integrability conditions, 2. nowhere zero mean curvature vector, the squared norm of the second fundamental form is bounded from above and some integrability conditions. Furthermore, I and Tomoya Miura showed that any triharmonic Riemannian submersion from a 3-dimensional space form is harmonic.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究ではEells-Sampsonにより導入された調和写像の一般化である2重調和写像，3重調和写像の中でも最も重要な問題であるBalmus-Montaldo-Oniciuc予想とChen's conjectureとその一般化の肯定的部分的解決を与えている。