Development of Theory of Harmonic Maps

2017 Fiscal Year Final Research Report

• Project/Area Number: 25400154
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: Urakawa Hajime 東北大学, 高度教養教育・学生支援機構, 名誉教授 (50022679)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: エネルギー汎関数 / ２－調和写像 / 調和写像 / ２－エネルギー汎関数 / 対称空間 / ２－調和部分多様体 / 葉層多様体 / コーシー・リーマン多様体

Outline of Final Research Achievements

Harmonic map is a critical point of the energy and biharmonic map is the one of the 2-energy. Both theories have been developed greatly in the world. We have clarified the similarities and differences between these theories, recently. We have constructed several examples of biharmonic maps which are not harmonic maps. We gave and clarified the conditions which biharmonic maps must be harmonic maps.
We have constructed and classified all homogeneous biharmonic submanifolds in the symmetric space which are not harmonic.
We gave the similar notions of biharmonic foliations and also pseudo biharmonic submanifolds in Cauchy-Riemannian geometry, and gave several such examples.

Free Research Field

微分幾何学及び大域解析学