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Long-time behavior of Ricci flow on some complex surfaces

John Lott

TL;DR

The paper analyzes the long-time behavior of the Kähler–Ricci flow on certain four-dimensional complex surfaces of general type that contain disjoint $(-2)$-curves. It builds biLipschitz model flows by gluing an approximate Eguchi–Hanson cap to the expanding flow on the canonical orbifold model $X$, proving that the Ricci flow stays close to the model with a convergence rate $K(t)=1+O(t^{-1+\epsilon})$; contracting the $(-2)$-curves yields a negative-Einstein orbifold, and in the complex-hyperbolic case the convergence improves to $K(t)=1+O(t^{-2+\epsilon})$ with a stability result. The construction relies on a two-parameter family of Eguchi–Hanson metrics, a detailed potential-flow reduction to a scalar equation, and a fixed-point argument in weighted Hölder spaces to control the gluing error. Overall, the work provides a precise, quantitative description of the hybrid expanding/static long-time behavior of these four-dimensional Ricci flows and extends the understanding of how singularities and ALE caps interact under the flow.

Abstract

We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.

Long-time behavior of Ricci flow on some complex surfaces

TL;DR

The paper analyzes the long-time behavior of the Kähler–Ricci flow on certain four-dimensional complex surfaces of general type that contain disjoint -curves. It builds biLipschitz model flows by gluing an approximate Eguchi–Hanson cap to the expanding flow on the canonical orbifold model , proving that the Ricci flow stays close to the model with a convergence rate ; contracting the -curves yields a negative-Einstein orbifold, and in the complex-hyperbolic case the convergence improves to with a stability result. The construction relies on a two-parameter family of Eguchi–Hanson metrics, a detailed potential-flow reduction to a scalar equation, and a fixed-point argument in weighted Hölder spaces to control the gluing error. Overall, the work provides a precise, quantitative description of the hybrid expanding/static long-time behavior of these four-dimensional Ricci flows and extends the understanding of how singularities and ALE caps interact under the flow.

Abstract

We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.
Paper Structure (8 sections, 10 theorems, 60 equations)

This paper contains 8 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Kähler-Ricci flow and potential flow
  4. Eguchi-Hanson metric
  5. Approximate Kähler-Ricci flow on the caps
  6. Model flow
  7. Proof of Theorem \ref{['thm1']}
  8. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let $M$ be a minimal complex surface of general type, with disjoint rational curves $\{E_i\}$ of self intersection $-2$. Let $[E_i] \in \operatorname{H}^{1,1}(M; {\mathbb R})$ be the cohomology class dual to the homology class of $E_i$. Given positive numbers $\{b_i\}$, there is a model flow of Kähl

Theorems & Definitions (19)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Example 1
  • ...and 9 more