No semistability at infinity for Calabi-Yau metrics asymptotic to cones

TL;DR

This paper proves a phenomenon where complete Calabi-Yau metrics asymptotic to cones do not exhibit semistability at infinity, contrasting with Kähler-Einstein singularities, and establishes a polynomial convergence rate to the asymptotic cone.

Contribution

It introduces a no semistability at infinity result for Calabi-Yau metrics asymptotic to cones, refining the understanding of their degeneration behavior.

Findings

Elimination of intermediate K-semistable cones in the degeneration theory.

Polynomial convergence rate to the asymptotic cone.

Contrast with Kähler-Einstein local singularities.

Abstract

We discover a "no semistability at infinity" phenomenon for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson and the first author. It is in sharp contrast to the setting of local singularities of K\"ahler-Einstein metrics. A byproduct of the proof is a polynomial convergence rate to the asymptotic cone for such manifolds, which bridges the gap between the general theory of Colding-Minicozzi and the classification results of Conlon-Hein.