Asymptotically conical Calabi–Yau manifolds, I

Abstract

This is the first of a series of articles on complete Calabi–Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition $O\left(r^{-n-\epsilon }\right)$ needed in earlier work to $O\left(r^{-\epsilon }\right)$, relying on some new ideas about harmonic functions. We then look at two classes of examples: crepant resolutions of cones (this includes a new class of Ricci-flat small resolutions associated with flag varieties) and affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on $T^{\ast }{S}^n$ is $-2(n/(n-1\left)\right)$.