Complexified diffeomorphism groups, totally real submanifolds and Kähler–Einstein geometry
Complexified diffeomorphism groups, totally real submanifolds and Kähler–Einstein geometry
复制标题
复杂的微分同胚群、完全真实的子流形和凯勒-爱因斯坦几何
DOI：
10.1090/tran/7421
复制
发表时间：
2015
影响因子：
1.3
通讯作者：
T. Pacini
中科院分区：
文献类型：
--
作者：
Jason D. Lotay;T. Pacini
设(M，J)是一个几乎复流形。我们证明了M中全实子流形的无限维空间Tau具有自然联系。这引出了Tau中测地线的规范概念，以及定义在Tau上的泛函何时是凸的相应定义。
Tau中的测地线可以表示为M中的J-全纯曲线族；我们证明了一个唯一性结果，并研究了它们的存在性。当M是Kahler时，我们在Tau上定义了一个正则泛函；如果M具有非正的Ricci曲率，则它是凸的。
我们的构造在形式上类似于Donaldson，Fujiki和Semmes研究的Kahler势空间上的测地线和Mabuchi泛函的概念。受此类比的启发，我们讨论了我们的理论在负卡勒-爱因斯坦流形中极小拉格朗日的研究中的可能应用。
Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex.
Geodesics in Tau can be expressed in terms of families of J-holomorphic curves in M; we prove a uniqueness result and study their existence. When M is Kahler we define a canonical functional on Tau; it is convex if M has non-positive Ricci curvature.
Our construction is formally analogous to the notion of geodesics and the Mabuchi functional on the space of Kahler potentials, as studied by Donaldson, Fujiki and Semmes. Motivated by this analogy, we discuss possible applications of our theory to the study of minimal Lagrangians in negative Kahler-Einstein manifolds.