Twistor spaces, Einstein metrics and isomonodromic deformations
The importance of twistor theory results from its ability to translate differential geometric questions into equivalent problems in algebraic geometry. For example, an anti-self-dual conformal structure on a four-manifold \(M\) can be encoded as the holomorphic structure of a complex three-manifold, its twistor space. In this paper, anti-self-dual manifolds \(M\) are studied, which carry an action of the Lie group SU(2), preserving the conformal structure and having three-dimensional orbits. The natural lifting of this action on the twistor space \(Z\) defines a homomorphism of holomorphic vector bundles \(\alpha:{\mathcal O}_Z\otimes {\mathfrak g}^c\to TZ\), whose rank is at least two. If the image sheaf of \(\alpha\) has rank two, then he shows the conformal structure on \(M\) to be either conformally flat, or the Eguchi-Hanson metric, or a member of the family described by \textit{V. A. Belinskiĭ, G. W. Gibbons, D. N. Page} and \textit{C. N. Pope} [Phys. Lett. B 76, No. 4, 433-435 (1978)]. The bulk of this paper is concerned with the case where \(\alpha\) has image of rank three. In this case, \(\alpha\) fails to be an isomorphism over an anticanonical divisor \(Y\subset Z\). The case where \(Y=2D\) is a multiple of a half-anticanonical divisor leads to Painlevé's third equation and is studied in a paper of \textit{H. Pedersen} and \textit{Y. S. Poon} [Classical Quantum Gravity 7, No. 10, 1707-1719 (1990; Zbl 0711.53039)]. The author studies in full detail the case where the divisor \(Y\) intersects a generic twistor line at four distinct points. On a generic twistor line, outside these four intersection points the morphism \(\alpha\) defines a flat connection whose holonomy representation does not depend on the twistor line. Such a family of connections is called an isomonodromic deformation. The residues of these connections on the four points of intersection define four trace-free \(2\times 2\) matrices \(A_i\), fulfilling the Schlesinger equation [see \textit{B. Malgrange}, Mathématique et physique, Sém. éc. Norm. Supér., Paris 1979-1982, Prog. Math. 37, 401-426 (1983; Zbl 0528.32017)]. He then gives an explicit formula for the conformal structure on \(M\) in terms of these matrices \(A_i\), under the condition that the metric can be put in diagonal form. If the conformal structure admits an Einstein metric, this can be achieved. Important for a complete description of the Einstein case will be the study of the monodromy group of the connection on the punctured twistor lines. Generalizing earlier results [the author, Symp. Math. 36, 190-222 (1996)] he shows that this group is almost Abelian; more precisely, it contains an Abelian subgroup of index two. This turns out to be the key to obtain explicit solutions, because the induced connection on the elliptic curve \(C\), defined as the double covering of a twistor line branched over the four singularities of the connection, will have Abelian holonomy. The monodromy of the pulled-back connection is therefore obtained by a calculation of periods of differentials on the curve \(C\). Using elliptic functions, a formula for the connection matrix on the twistor lines is then obtained. Writing the connection matrix in a specific form, he obtains a single function \(y\) such that Schlesinger's equation for the connection is equivalent to Painlevé's sixth equation for \(y\). In this way, the isomonodromic deformation explicitly constructed before, yields a solution of a particular case of Painlevé VI. This function is written explicitly using theta functions. The author uses this function to give an explicit formula for a family of complete anti-self-dual Einstein metrics on the open four-ball whose conformal structure extends across the boundary. To find the metric within the conformal structure, he uses a formula of \textit{K. P. Tod} [Phys. Lett. A 190, No. 3-4, 221-224 (1994)]. Putting together his results, he obtains a complete classification of anti-self-dual SU(2)-invariant Einstein metrics with three-dimensional orbits. Besides the explicit solution of a special class of Painlevé VI and the description of anti-self-dual metrics, the main achievement of this paper is the result that in the case of anti-self-dual conformal structures with SU(2)-symmetry, the conformal structure is encoded not by holomorphic but by topological or algebraic data, namely the representation of a fundamental group.
scientific article; zbMATH DE number 844214