Gluing Eguchi-Hanson Metrics and a Question of Page

In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kahler 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 that 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 that are neither Kahler nor self-dual. In this paper, we focus on a configuration of maximal symmetry involving eight small Eguchi-Hanson manifolds of each orientation that are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson manifolds with opposite orientation, we identify a nonvanishing 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)1/2, while the maximum of the Ricci curvature converges to 0.(c) 2016 Wiley Periodicals, Inc.