Warped quasi-asymptotically conical Calabi-Yau metrics

TL;DR

This paper constructs new complete Calabi-Yau metrics with maximal volume growth on smoothings of Calabi-Yau cones, analyzing their geometry at infinity and demonstrating transition behaviors between different cones.

Contribution

It introduces a method to build and analyze Calabi-Yau metrics with complex asymptotic structures, including metrics with conical singularities and non-unique tangent cones.

Findings

Constructed new examples of complete Calabi-Yau metrics with maximal volume growth.

Described the geometry at infinity using compactifications and weighted blow-ups.

Showed existence of metrics with transition behavior between different Calabi-Yau cones.

Abstract

We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted H\"older spaces. Our methods also produce singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, in particular yielding a transition behavior between different Calabi-Yau cones as conjectured by Yang Li. This is used to exhibit many examples where the tangent cone at infinity does not uniquely specify a complete Calabi-Yau metric with exact K\"ahler form.