The analogy between self-dual connections and self-dual conformal structures is a close one. Indeed, the origins of this area of study put both topics together - they become, via twistor theory, holomorphic bundles and complex three-manifolds. However, the “grafting” procedure of Taubes which allows one to construct self-dual connections by adding in an extra instanton is far more difficult to achieve in the more non- linear realm of self-dual conformal structures. The question one asks is whether the connected sum of two self-dual conformal 4-manifolds can be given a self-dual conformal structure. Using twistor techniques, S. Donaldson and R. Friedman [Nonlinearity 2, No. 2, 197-239 (1989; Zbl 0671.53029)] succeeded in proving theorems of this type. The author here adopts an analytical approach to prove the existence for a connected sum of complex projective planes. In fact, work of C. LeBrun, building on that of Poon, gives some explicit conformal structures on such a connected sum [ J. Differ. Geom. 34, 223-253 (1991; Zbl 0711.53055)].
The author’s own approach rests on the study of self-dual structures on “cylindrical charts” \({\mathbb{R}}\times S^ 3\), a suitable Fredholm theory, and a glueing procedure. Characteristically, the equations on the cylinder can be viewed as an ordinary differential equation on the set of metrics on the sphere just as self-dual connections are viewed in Floer theory as the gradient flow of the Chern-Simons functional. It should be noted that in a recent preprint, Taubes has carried out an analytical approach to the same problem which yields the remarkable result that any 4-manifold, after taking the connected sum with a sufficiently large number of \(\mathbb{C} \mathbb{P}^ 2\)’s, admits a self-dual conformal structure.