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An analytic construction of dihedral ALF gravitational instantons

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Metric and Analytic Aspects of Moduli Spaces

Gravitational instantons are 4-dimensional complete non-compact hyperkhler manifolds with some curvature decay at infinity. The asymptotic geometry of these spaces plays an important role in a conjectural classification; for example, instantons of euclidean, i.e. quartic, large ball volume growth, are completely classified by Kronheimer, whereas the cubic regime, i.e. the {it ALF (Asymptotically Locally Flat)} case, is not fully understood yet. More precisely, ALF instantons with {it cyclic topology at infinity} are classified by Minerbe; by contrast, a classification in the {it dihedral} case at infinity is still unknown.

A wide, conjecturally exhaustive, range of dihedral ALF instantons were constructed by Cherkis-Kapustin, adopting the moduli space point of view, and studied explicitly by Cherkis-Hitchin. I shall explain in this talk another construction of such spaces, based on the resolution of a Monge-Ampre equation in ALF geometry.

This talk is part of the Isaac Newton Institute Seminar Series series.

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