Twistor 空间及相关复几何问题

The twistor construction is an important technique in differential geometry and mathematical physics. Given an even dimensional oriented Riemannian manifold, its twistor space has two canonical almost complex structures, introduced by Atiyah-Hitchin-Singer and Eells-Salamon, respectively. In 2011, G. Deschamps constructed many almost complex structures on the twistor space. Based on some special metrics (such as Balanced metrics, pluriclosed metrics, generalized Gauduchon metrics) introduced in complex geometry, The following questions will be thoroughly studied: (1) Studying the special twistor geometry of Riemannian 4-manifolds by using the method of moving frames, and considering the existence problems of pluriclosed metrics and 1-Gauduchon metrics on the twistor spaces; (2) Studying the special twistor geometry of general even dimensional Riemannian manifolds, especially the special twistor geometry of Quaternionic Kähler manifolds, and considering the existence problems of pluriclosed metrics and generalized Gauduchon metrics on the twistor spaces.

Twistor构造是微分几何与数学物理中的一项重要技术。对于给定的偶数维定向黎曼流形，其Twistor空间上有两个典型的近复结构，分别由Atiyah-Hitchin-Singer与Eells-Salamon引入的。在2011年，G. Deschamps在Twistor空间上引入了更多的近复结构。基于复几何中提出的一些特殊度量的概念（比如说：Balanced度量、pluriclosed 度量、广义Gauduchon度量），我们将深入研究如下问题：（1）通过活动标架法，研究四维黎曼流形的特殊Twistor几何，主要考虑pluriclosed 度量与1-Gauduchon度量的存在性问题；（2）研究一般偶数维黎曼流形的特殊Twistor几何，特别是Quaternionic Kähler流形的特殊Twistor几何，主要考虑pluriclosed 度量与广义Gauduchon度量的存在性问题。