Study of the Rumin complex of a CR manifold and the quaternionic CR manifold

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610009
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: Nayatani Shin 名古屋大学, 多元数理科学研究科, 教授 (70222180)
• Co-Investigator(Renkei-kenkyūsha): KAMADA HIROYUKI 宮城教育大学, 教育学部, 教授
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 強擬凸CR多様体 / Rumin複体 / ボホナー・ワイツェンベック型公式 / 四元数CR多様体 / ツイスター空間

Outline of Final Research Achievements

We studied sharp Bochiner-Weitsenboeck formulas for the Rumin complex of a strongly pseudoconvex CR manifold. For 1-forms we verified that the formula formerly written down by Rumin was actually sharp, and as an application we obtained a quite simple new proof of the sharp estimate of the first nonzero eigenvalue of the sublaplacian. Also, for 2-forms we tried to write down a sharp formula and determined it up to a single real parameter.
We studied the twistor space of a quaternionic CR manifold in order to define an (integrable) CR structure on it, and proved that one could define an almost CR structure on the twistor space and it was partially integrable.

Free Research Field

微分幾何学