The asymptotic behavior of the steady gradient K\"ahler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

TL;DR

This paper analyzes the asymptotic structure of a specific steady gradient K"ahler-Ricci soliton of Taub-NUT type, showing it is an ALF Calabi-Yau metric and constructing new examples using Tian-Yau-Hein methods.

Contribution

It determines the asymptotic cone of the soliton and constructs new ALF Calabi-Yau metrics on crepant resolutions of its quotients.

Findings

Identified the asymptotic cone of the Taub-NUT type soliton.

Proved the soliton is an ALF Calabi-Yau metric in a special case.

Constructed new ALF Calabi-Yau metrics on crepant resolutions.

Abstract

We first determine the asymptotic cone of the steady gradient K\"ahler-Ricci soliton of the Taub-NUT type constructed by Apostolov and Cifarell. Then we study a special case and prove that it is an ALF Calabi-Yau metric in a certain sense. Finally we construct new ALF Calabi-Yau metrics on crepant resolution of its quotients modeled on it using the method of Tian-Yau-Hein.