Gluing Eguchi-Hanson metrics and a question of Page

Abstract

In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat K\"ahler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if successful, this construction would produce Einstein metrics which are neither K\"ahler nor self-dual. In this paper, we focus on a configuration of maximal symmetry involving $8$ small Eguchi-Hanson manifolds of each orientation which are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson manifolds with opposite orientation, we identify a non-vanishing obstruction to the gluing problem, thereby destroying any hope of producing a metric of zero Ricci curvature in this way. Using this obstruction, we are able to understand the dynamics of such metrics under Ricci flow as long as the Eguchi-Hanson manifolds remain small. In particular, for the configuration described above, we obtain an ancient solution to the Ricci flow with the property that the maximum of the Riemann curvature tensor blows up at a rate of $(-t)^{\frac{1}{2}}$, while the maximum of the Ricci curvature converges to $0$.