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The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

Daheng Min

Abstract

We first determine the asymptotic cone of the steady gradient Kähler-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.

The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

Abstract

We first determine the asymptotic cone of the steady gradient Kähler-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.

Paper Structure

This paper contains 17 sections, 39 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. The steady gradient Kähler-Ricci soliton
  3. The locally flat metric
  4. Change of variable for $g^\prime_{\theta}$
  5. Change of variable for $g_{\xi}^\prime$
  6. Locally flatness of $g^\prime$
  7. The asymptotic cone of the locally flat metric
  8. The asymptotic cone of the Kähler-Ricci soliton
  9. The special case where $l=2$ and $a=0$
  10. Description of the metric
  11. The $SOB(2n-1)$ property
  12. A Kähler potential
  13. Estimation of the Riemannian curvature
  14. ALF Calabi-Yau metrics modeled on the metric of Apostolov-Cifarelli
  15. Construction of an asymptotic Calabi-Yau metric
  16. ...and 2 more sections

Key Result

Theorem 1.1

The asymptotic cone of the Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli is unique and is $(\prod_{j=1}^{l-1}\mathbb{C}^{d_j+1}/\Lambda)\times \mathbb{R}$. Where $\Lambda$ is a closed subgroup of $\mathbb{T}^{l-1}$ that acts on $\prod_{j=1}^{l-1}\mathbb{C}^{d_j+1}$, and depend

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 68 more