Asymptotically conical Calabi-Yau manifolds, I

TL;DR

This paper establishes existence and uniqueness results for complete Calabi-Yau manifolds asymptotic to cones, introduces new decay conditions, and analyzes specific examples including crepant resolutions and affine deformations.

Contribution

It proves general existence and uniqueness theorems for asymptotically conical Calabi-Yau manifolds with relaxed decay conditions and examines explicit examples and decay rates.

Findings

Uniqueness results hold under weaker decay conditions.

Constructs new Ricci-flat resolutions related to flag manifolds.

Determines optimal decay rate for the Stenzel metric.

Abstract

This is the first part in a two-part series 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 $O(r^{-n-\epsilon})$ needed in earlier work to $O(r^{-\epsilon})$, relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) 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 $T^*S^n$ is $-2\frac{n}{n-1}$.