Celebratio Mathematica — Atiyah

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M. F. Atiyah: “Green’s functions for self-dual four-manifolds,” pp. 129–158 in Mathematical analysis and applications, Part A, published as Adv. in Math. Suppl. Stud. 7A. Issue edited by L. Nachbin. Academic Press (New York), 1981. MR 634238 Zbl 0484.53036 incollection

The Penrose twistor theory converts problems of four-dimensional Riemannian geometry into complex analytic geometry of three dimensions. It applies whenever the Riemannian curvature is self-dual, in the sense that the Weyl tensor (the conformally invariant part of the Riemannian curvature) is self-dual. For a general exposition of this theory (in the positive definite case) see [Atiyah, Hitchin and Singer 1978]. Now on a Riemannian four-manifold there is a conformally invariant version of the Laplace operator (obtained by adding one-sixth of the scalar curvature of the usual Laplacian) and this has therefore a conformally invariant Green’s function (under appropriate global hypotheses). The purpose of this chapter is to show what corresponds to this Green’s function in the twistor theory. It turns out that we get a very well-known object in complex analytic geometry associated to the diagonal in \( Z \times Z \), where \( Z \) is the twistor space.