Equivariant connected sums of compact self-dual manifolds
Equivariant connected sums of compact self-dual manifolds
复制标题
紧致自对偶流形的等变连通和
DOI：
10.1007/bf01446656
复制
发表时间：
1995
影响因子：
1.4
通讯作者：
Y. Poon
中科院分区：
文献类型：
--
作者：
H. Pedersen;Y. Poon
由于最近的陶伯定理，紧凑的四维流形上的自偶联结构存在于丰度中[43]。因此，在这个主题中，通常不再找到示例是一个问题。为了加深我们对自我偶然形成几何形状的理解，我们采取了试图理解这种几何形状可能对称群体的道路。即保持保形转换的方向组。勒布朗的双曲线ansatz是第一种系统的方法，该方法与非平凡的对称性组构造了大量紧凑的自动歧管[22]。 ANSATZ的条件是对称组包含一个S l：= u（1），该s = u（1）作用于歧管半且半纯的，这意味着歧管的任何点的各向同性组都是微不足道或整个组S I。由于这种条件在小组动作和对基础歧管的某些有利的拓扑条件下，给定的四维流形上的几何形式的自偶性方程将减少到S双曲线3空间上的1个孔路方程。通过这种设置，Lebrun明确构建了大量的紧凑自动二线，具有半无s L_SMEMEMEMETRY，包括复杂的射击平面CI上的fubinistudy指标？ 2和一个在[38]中首次发现的C〜＃C〜的自偶像指标的单参数家族。我们将NCI上的自偶像指标称为？ 2，n> _ 1，由Lebrun的双曲线Ansatz（Lebrun指标）构建。在这个非常成功的安萨兹之后，很自然地尝试找到超出该方法范围的新示例。
Due to a recent theorem of Taubes, self-dual conformal structures on compact four-dimensional manifolds exist in abundance [43]. Therefore, finding examples in general is no longer an issue in this subject. To deepen our understanding of self-dual conformal geometry, we take the path of trying to understand the possible symmetry groups of this geometry; i.e. the groups of orientation preserving conformal transformations. The first systematic approach which constructs a large collection of compact self-dual manifolds with a non-trivial group of symmetries is LeBrun's hyperbolic Ansatz [22]. The condition of this Ansatz is that the symmetry group contains an S l := U(1) which acts on the manifold semi-freely, meaning that the isotropy group at any point of the manifold is either trivial or the entire group S I. With this condition on the group action and some favourable topological conditions on the underlying manifold, the geometrical equations of self-duality on the given four-dimensional manifold are reduced to the S 1-monopole equations on hyperbolic 3-space. With this setup, LeBrun explicitly constructed a large number of compact self-dual manifolds with semi-free S l_symmetry including the FubiniStudy metric on the complex projective plane CI? 2 and a one-parameter family of self-dual metrics on C ~ # C ~ first found in [38]. We shall call the self-dual metrics on nCI? 2, n >_ 1, constructed by LeBrun's hyperbolic Ansatz, the LeBrun metrics. After this very successful Ansatz, it is natural to try to find new examples which are beyond the scope of that method.