- qc / 0 30 40 66 v 3 3 1 Ju l 2 00 3 Anti - self - dual Riemannian metrics without Killing vectors , can they be realized on K 3 ?

Explicit Riemannian metrics with Euclidean signature and anti-self dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogenous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on K3, or on surfaces whose universal covering is K3. PACS numbers: 04.20.Jb, 02.40.Ky 2000 Mathematics Subject Classification: 35Q75, 83C15 We present Riemannian metrics with anti-self-dual curvature that admit no Killing vectors. Our motivation to study this problem has been K3 which is the most important gravitational instanton [1]-[3]. It is necessary that the metric on K3 should not admit any continuous symmetries. Our solutions do satisfy this criterion which is necessary but not sufficient since K3 is a compact 4-dimensional Riemannian manifold. In this note all our considerations will be local and we shall not discuss the global problem of compactness. The metric ds = uik̄ dζ dζ̄ (1) must be hyper-Kähler. 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-