ar X iv : g r - qc / 0 30 40 66 v 1 1 8 A pr 2 00 3 Towards the metric on K 3

An explicit Riemannian metric with Euclidean signature and anti-self dual curvature that does not admit any Killing vectors is presented. It could be a class of metrics on K3, or on surfaces whose universal covering is K3. This metric contains an infinite number of arbitrary parameters, so further work is required to identify the finite number of essential parameters characterizing K3. PACS numbers: 04.20.Jb, 02.40.Ky 2000 Mathematics Subject Classification: 35Q75, 83C15 The most important gravitational instanton is K3 [1]-[3]. This is a compact 4-dimensional Riemannian manifold with anti-self-dual curvature. Its metric ds = uik̄ dζ dζ̄ (1) is hyper-Kähler with vanishing first Chern class [4]. Therefore it satisfies the Euclidean Einstein field equations. It has been over a century since Kummer [5] introduced K3 as a quartic surface in CP 3 and half a century since Calabi [6] pointed out that the Kähler potential satisfies the elliptic complex Monge-Ampère equation u11̄u22̄ − u12̄u1̄2 = 1 (2) hereafter to be referred to as CMA2. Yau [7] has given an existence and uniqueness proof but so far there are no explicit exact solutions of CMA2 appropriate to K3. Another testament to the difficulties encountered in dealing with complex Monge-Ampère equations lies in the fact that its homogeneous version replaces Laplace’s equation as the fundamental equation governing 1 functions of many complex variables [8]. The principal difficulty in constructing solutions of CMA2 that would describe K3 lies in the requirement that the Kähler metric (1) must not admit any Killing vectors. In the language of differential equations such solutions are known as non-invariant solutions of (2). Recently we suggested that the method of group foliation [9, 10, 11] can serve as a regular tool for finding non-invariant solutions of PDEs. Group foliation was carried out for CMA2 and the Boyer-Finley equations in [12] and [13] respectively using their infinite symmetry groups [14]. However, in this note we shall adopt a different approach which turned out to be fruitful specifically for CMA2, to find an explicit metric without any Killing vectors that has anti-self-dual curvature. We emphasize at the outset that the class of solutions we are considering here is not the full set of solutions of CMA2. In our approach we start with the Mason-Woodhouse [15] Lax pair and supplement the Lax equations with two more linear equations such that CMA2 emerges as an algebraic compatibility condition. The would be Baker-Akhiezer function in the standard Lax approach is now regarded as a complex potential. Choosing symmetry characteristics [16] of CMA2 for the real and imaginary parts of this potential we arrive at an over-determined set of linear equations satisfied by one real potential. This system is the image of CMA2 supplemented by some differential constraints after performing a Legendre transformation. It is therefore also the image of a particular set of solutions of the Einstein equations with anti-self-dual Riemann curvature 2-form. Here we shall present only the final results and postpone the detailed derivation to a future publication, a preliminary account of which can be found in [17]. We use the Euclidean Newman-Penrose formalism [18], [19] to write the metric in the form ds = l ⊗ l̄ + l̄ ⊗ l +m⊗ m̄+ m̄⊗m (3) where the co-frame ω = {l, l̄, m, m̄} is given by l = 1 v [C(C − |A|2)] [ C(Cdz +Bdz) + Ā(Cdz̄ + B̄dz̄) ]