Daheng Min | Searcharxiv

arXiv2024

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

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.

math.DG↗

arXiv2023

Construction of higher-dimensional ALF Calabi-Yau metrics

Roughly speaking, an ALF metric of real dimension 4n should be a metric such that it has a (4n-1)-dimensional asymptotic cone, the volume growth of this metric is of order 4n-1 and its sectional curvature tends to 0 at infinity. In this paper, we first show that the Taub-NUT deformation of a hyperk\"ahler cone with respect to a locally free S1-symmetry is ALF hyperk\"ahler. Using this metric at infinity, we establish the existence of ALF Calabi-Yau metric on certain crepant resolutions. In particular, we prove that there exist ALF Calabi-Yau metrics on canonical bundles of classical homogeneous Fano contact manifolds.

math.DG↗