Gluing theorems for complete anti-self-dual spaces

Abstract

1.1 Summary. One of the special features of 4-dimensional differential geometry is the existence of objects with self-dual (SD) or anti-self-dual (ASD) curvature. The objects in question can be connections in an auxiliary bundle over a 4-manifold, leading to the study of instantons in Yang– Mills theory [DK], or as in this paper, Riemannian metrics or conformal structures. Although such ASD conformal structures give absolute minima of the functional c �→ � W (c)� 2 ,w hereW (c) denotes the Weyl tensor of the conformal structure c, variational methods are not well suited to the study of this problem, essentially because of its conformal invariance. For this reason, gluing theorems provide a very important source of information about ASD conformal structures. Our purpose in this paper is to give some new and rather general gluing theorems for ASD and HermitianASD conformal structures, following the method suggested by Floer in [F]. The prototypical gluing theorem takes a pair (Xj ,c j )( j =1 , 2) of compact conformally ASD 4-manifolds and analyzes the problem of finding an ASD conformal structure c on X = X1�X 2 that is ‘close to’ cj in suitable subsets Xj\Bj ⊂ X1�X 2. In this situation there exist finite-dimensional vector spaces (the obstruction spaces) H 2 cj (Xj) whose vanishing is sufficient to guarantee the existence of c with the desired properties. (If H 2 cj (Xj) � , then the gluing theorem yields a map from another finite-dimensional vector space into H 2 c1 (X1) ⊕ H 2 c2 (X2), the zeroes of which yield ASD conformal structures on X1�X 2.) The result just stated (gluing for compact conformally ASD spaces) was proved by Donaldson and Friedman [DF] and in a very special case by Floer [F]. The approach of [DF] was to exploit the twistor description [P], [AtHS] of conformally ASD spaces, to translate the gluing problem into one of deformation theory of complex singular spaces. Floer, on the other hand, worked directly with the 4-manifolds and used some tools from the theory of elliptic operators on non-compact manifolds with cylindrical ends.