Equivariant connected sums of compact self-dual manifolds

Abstract

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.