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On Kähler Ricci shrinker surfaces

Yu Li, Bing Wang

TL;DR

This paper proves that every Kähler Ricci shrinker surface has bounded scalar curvature, and thus bounded sectional curvature, removing previously needed curvature assumptions. The authors develop a two-part canonical neighborhood theory to analyze high-curvature regions: part A shows convergence to steady soliton orbifolds, and part B shows convergence to the cylindrical ancient Kähler Ricci flow on $\\mathbb{P}^1 \\times \\mathbb{C}$; both results rely on a refined $\\mathbb{F}$-compactness framework, local scalar-curvature estimates, and splitting arguments using the potential function $f$. Combining these tools with existing classification results, they obtain a complete list of Kähler Ricci shrinker surfaces: the Gaussian soliton on $\\mathbb{C}^2$, the Feldman-Ilmanen-Knopf (FIK) shrinker on the blowup of $\\mathbb{C}^2$, the standard product $\\mathbb{P}^1\\times\\mathbb{C}$, the Bamler-Cifarelli-Conlon-Deruelle (BCCD) shrinker on the blowup of $\\mathbb{P}^1\\times\\mathbb{C}$, and, in the closed case, del Pezzo surfaces with unique shrinker metrics. The results significantly advance the understanding of four-dimensional Kähler shrinkers without curvature bounds and unify several prior constructions under a single framework.

Abstract

In this paper, we prove that any Kähler Ricci shrinker surface has bounded sectional curvature. Combining this estimate with earlier work by many authors, we provide a complete classification of all Kähler Ricci shrinker surfaces.

On Kähler Ricci shrinker surfaces

TL;DR

This paper proves that every Kähler Ricci shrinker surface has bounded scalar curvature, and thus bounded sectional curvature, removing previously needed curvature assumptions. The authors develop a two-part canonical neighborhood theory to analyze high-curvature regions: part A shows convergence to steady soliton orbifolds, and part B shows convergence to the cylindrical ancient Kähler Ricci flow on ; both results rely on a refined -compactness framework, local scalar-curvature estimates, and splitting arguments using the potential function . Combining these tools with existing classification results, they obtain a complete list of Kähler Ricci shrinker surfaces: the Gaussian soliton on , the Feldman-Ilmanen-Knopf (FIK) shrinker on the blowup of , the standard product , the Bamler-Cifarelli-Conlon-Deruelle (BCCD) shrinker on the blowup of , and, in the closed case, del Pezzo surfaces with unique shrinker metrics. The results significantly advance the understanding of four-dimensional Kähler shrinkers without curvature bounds and unify several prior constructions under a single framework.

Abstract

In this paper, we prove that any Kähler Ricci shrinker surface has bounded sectional curvature. Combining this estimate with earlier work by many authors, we provide a complete classification of all Kähler Ricci shrinker surfaces.
Paper Structure (6 sections, 23 theorems, 164 equations, 3 figures, 1 table)

This paper contains 6 sections, 23 theorems, 164 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical neighborhood theorem—part A
  4. Canonical neighborhood theorem—part B
  5. Proof of the main theorem
  6. Metric flows and $\mathbb F$-convergence

Key Result

Theorem 1.1

Let $(M^2,g,J,f)$ be a Kähler Ricci shrinker surface. Then $(M^2,g,J,f)$ is biholomorphic-isometric to one of the following manifolds.

Figures (3)

  • Figure 1: $\mathcal{R}'_{t_j}\cong S^2 \times \mathbb{R}$
  • Figure 2: $\mathcal{R}'_{t_j}\cong S^2 \times S^1$
  • Figure 3: Case (iii) or Case (iv) holds for all components

Theorems & Definitions (51)

  • Theorem 1.1: Complete classification of Kähler Ricci shrinker surfaces
  • Theorem 1.2: Main result
  • Theorem 1.3: Canonical Neighborhood Theorem
  • proof : Outline of proof of Theorem \ref{['T101']}
  • proof : Outline of proof of Theorem \ref{['T102']}
  • Lemma 2.1: CZ10 HM11
  • Definition 2.2
  • Example 2.3: Standard model
  • Definition 2.4: $\epsilon$-close to a pointed Kähler Ricci flow
  • Definition 3.1
  • ...and 41 more