May 27, 2021
TITLE: Gluing Eguchi-Hanson Metrics and a Question of Page
ABSTRACT:
I will discuss a paper of Simon Brendle and myself (Comm. Pure Appl. Math. 70 (2017), no. 7, 1366-1401) motivated by a question Page asked in 1981. Page’s question was based on a physical picture for the Ricci-flat Kähler metrics on the K3 surface proposed by Gibbons-Pope and Page in 1978. In this picture the K3 metrics are viewed as desingularizations by Eguchi-Hanson manifolds of the 16 orbifold points of the quotient of a torus by the antipodal map. Such a construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. Page’s question was whether some of the Eguchi-Hanson metrics can be attached with the opposite orientation resulting in a manifold of different topology carrying a non-K\”ahler Ricci-flat metric.
We imposed a discrete group of symmetries on such a construction to simplify the situation while still capturing the essential features. We then studied the obstructions to such a construction which arise from the interaction of the Eguchi-Hanson manifolds being attached, because the obstructions arising from the interaction with the flat background all vanish. It turns out that these obstructions cannot be overcome and the construction fails. Finally we made use of the non-vanishing obstruction to construct and study ancient solutions of the Ricci flow where the Eguchi-Hanson manifolds attached shrink to orbifold points as 
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