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A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

Richard H. Bamler, Eric Chen, Ronan J. Conlon

Abstract

We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler--Ricci soliton.

A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

Abstract

We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler--Ricci soliton.

Paper Structure

This paper contains 12 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Result
  4. Acknowledgements
  5. Preliminaries
  6. Riemannian cones
  7. Kähler cones
  8. Type II deformations of Kähler cones
  9. Asymptotically conical expanding Ricci solitons
  10. Asymptotically conical expanding Kähler--Ricci solitons
  11. Expanding solitons and the Ricci flow
  12. Proof of Theorem \ref{['thm_nonkahler']}

Key Result

Theorem 1

There exists a real four-dimensional complete non-Kähler expanding gradient Ricci soliton with one end with quadratic curvature decay with derivatives (or equivalently, AC) that has a Kähler tangent cone at infinity.

Theorems & Definitions (10)

  • Theorem 1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Claim 3.1
  • proof : Proof of Claim \ref{['perturb']}
  • Claim 3.2
  • proof : Proof of Claim \ref{['ronanan']}