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Stability of asymptotically conical gradient Kähler-Ricci expanders

Longteng Chen

TL;DR

The paper studies the stability of the normalized Kähler-Ricci flow on non-compact manifolds admitting an asymptotically conical gradient expanding Kähler-Ricci soliton. It reduces the flow to a complex Monge-Ampère equation using the Killing field $JX$ and applies a parabolic maximum principle to obtain global existence and uniform estimates on space-time. Under Condition II, the flow exists for all time and converges smoothly to an asymptotically conical gradient Kähler-Ricci expander; if the initial perturbation satisfies a quantitative decay, the limit coincides with the original soliton. The results extend stability phenomena known for Kähler-Einstein and compact cases to AC expanders, highlighting the role of AC geometry and soliton symmetries in governing long-time behavior. This contributes to understanding the moduli of AC solitons and the asymptotic structure of noncompact Kähler-Ricci flows with soliton symmetries.

Abstract

In this work, we consider a perturbation of an asymptotically conical gradient expanding Kähler-Ricci soliton metric $g$ in the same Kähler class. We demonstrate that, under suitable assumptions, the normalized Kähler-Ricci flow starting from the initial perturbed metric exists for all time and converges uniformly to an asymptotically conical gradient expanding Kähler-Ricci soliton metric $g_\infty$. Moreover, if the perturbed initial metric is asymptotic to $g$ at spatial infinity, then the limiting metric coincides with the original soliton, that is, $g_\infty = g$.

Stability of asymptotically conical gradient Kähler-Ricci expanders

TL;DR

The paper studies the stability of the normalized Kähler-Ricci flow on non-compact manifolds admitting an asymptotically conical gradient expanding Kähler-Ricci soliton. It reduces the flow to a complex Monge-Ampère equation using the Killing field and applies a parabolic maximum principle to obtain global existence and uniform estimates on space-time. Under Condition II, the flow exists for all time and converges smoothly to an asymptotically conical gradient Kähler-Ricci expander; if the initial perturbation satisfies a quantitative decay, the limit coincides with the original soliton. The results extend stability phenomena known for Kähler-Einstein and compact cases to AC expanders, highlighting the role of AC geometry and soliton symmetries in governing long-time behavior. This contributes to understanding the moduli of AC solitons and the asymptotic structure of noncompact Kähler-Ricci flows with soliton symmetries.

Abstract

In this work, we consider a perturbation of an asymptotically conical gradient expanding Kähler-Ricci soliton metric in the same Kähler class. We demonstrate that, under suitable assumptions, the normalized Kähler-Ricci flow starting from the initial perturbed metric exists for all time and converges uniformly to an asymptotically conical gradient expanding Kähler-Ricci soliton metric . Moreover, if the perturbed initial metric is asymptotic to at spatial infinity, then the limiting metric coincides with the original soliton, that is, .

Paper Structure

This paper contains 24 sections, 63 theorems, 355 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Statement of main theorems
  5. Initial conditions
  6. Stability of Kähler-Ricci flow on Kähler-Einstein manifolds
  7. Strategy of proof
  8. Outline of proof
  9. Acknowledgment
  10. Preliminaries
  11. Self-similar solutions and normalized Kähler-Ricci flow
  12. Asymptotic conical gradient Kähler-Ricci expanders and their structure theorem
  13. Geometric interpretation of Condition I and II
  14. Shi's solution
  15. Shi's solution to Ricci flow
  16. ...and 9 more sections

Key Result

Theorem 1

Let $(M, g, X)$ be an asymptotically conical gradient Kähler-Ricci expander with asymptotic Kähler cone $(C,g_0)$. Let $f$ be the normalized soliton potential as in Lemma soliton indentities. Assume that $\psi_0$ is a smooth real-valued function on $M$ satisfying Condition II (see Definition conditi

Theorems & Definitions (138)

  • Definition 1.1: Asymptotically conical gradient Kähler-Ricci expander
  • Theorem 1: Convergence theorem
  • Remark 1.2
  • Remark 1.3
  • Corollary 2
  • Theorem 3: Long time existence theorem
  • Remark 1.4
  • Definition 1.5: Condition I
  • Definition 1.6: Condition II
  • Remark 1.7
  • ...and 128 more