Connected Sums of Self-dual Manifolds and Equivariant Relative Smoothings

Recall that a conformal 4-manifold is called self-dual if its Weyl curvature, considered as a bundle valued 2-form, is in the +1 eigenspace of the Hodge star-operator [1]. Due to Schoen’s proof [19] of the Yamabe conjecture it is known that within any conformal class on a compact manifold is a metric whose scalar curvature is constant and the sign of this constant is a conformal invariant. The main objective of this paper is to ensure that the scalar curvature is positive for the self-dual structures on the connected sums nCP of complex projective planes found in [13]. The metrics admit a torus T 2 of orientation preserving conformal isometries. For self-dual metrics the total space Z of the bundle of anti-self-dual 2forms of unit length is a complex 3-manifold. This complex manifold is the twistor space [1] and it gives an alternative description of self-duality. Indeed, Donaldson and Friedman [2] used a desingularisation of a singular model of the desired twistor space to prove existence of self-dual structures on nCP. The self-dual metric on CP is the Fubini-Study metric and the full moduli on 2CP had previously been obtained [15] via a different twistor construction. In [13] we adapted the theory of Donaldson and Friedman to obtain equivariant connected sums of compact self-dual manifolds. If the symmetry group is at least three-dimensional it is known [16] that the conformal metric is of non-negative type. In contrast Kim [7] obtained S1-symmetric examples of negative scalar curvature while LeBrun [11] gave examples on nCP of positive scalar curvature and with symmetry group S1. These metrics were obtained via an ansatz involving monopoles on hyperbolic 3-space. Similarly, Joyce [6] obtained T 2-symmetric metrics on nCP of positive type using hyperbolic monopoles in two dimensions. These constructions give relatively easy access to knowledge about scalar curvature