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Calabi flow with bounded $L^p$ scalar curvature (II)

Haozhao Li, Linwei Zhang

TL;DR

The paper proves that the Calabi flow on a compact Kähler manifold can be extended beyond a finite time provided a space-time $L^p$-type bound on the scalar curvature and its Laplacian holds with $p>n$. It leverages the parabolic structure of the flow to establish a sequence of a priori estimates, starting with L∞ control of the potential and the auxiliary function $F$, then obtaining space-time $L^s$ bounds for $n+\Delta_g\varphi$ and gradient bounds for $\nabla F$ and $\nabla\varphi$, and finally bounding $\|n+\Delta_g\varphi\|_{\infty}$. These bounds feed into parabolic Hölder regularity for $F$ and $\varphi$, enabling an extension past $T$ via a standard continuation argument. The results connect Calabi flow extension criteria to analogous results for Ricci and mean curvature flows, highlighting the role of space-time curvature control in fourth-order geometric flows. The techniques combine Chen-Cheng parabolic estimates with parabolic Moser iteration and parabolic Sobolev inequalities on evolving Kähler metrics to achieve a comprehensive a priori estimate framework.

Abstract

In this paper, we show that on a compact Kähler manifold the Calabi flow can be extended as long as some space-time $L^p$ integrals of the scalar curvature are bounded.

Calabi flow with bounded $L^p$ scalar curvature (II)

TL;DR

The paper proves that the Calabi flow on a compact Kähler manifold can be extended beyond a finite time provided a space-time -type bound on the scalar curvature and its Laplacian holds with . It leverages the parabolic structure of the flow to establish a sequence of a priori estimates, starting with L∞ control of the potential and the auxiliary function , then obtaining space-time bounds for and gradient bounds for and , and finally bounding . These bounds feed into parabolic Hölder regularity for and , enabling an extension past via a standard continuation argument. The results connect Calabi flow extension criteria to analogous results for Ricci and mean curvature flows, highlighting the role of space-time curvature control in fourth-order geometric flows. The techniques combine Chen-Cheng parabolic estimates with parabolic Moser iteration and parabolic Sobolev inequalities on evolving Kähler metrics to achieve a comprehensive a priori estimate framework.

Abstract

In this paper, we show that on a compact Kähler manifold the Calabi flow can be extended as long as some space-time integrals of the scalar curvature are bounded.
Paper Structure (11 sections, 23 theorems, 186 equations)

This paper contains 11 sections, 23 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Estimates
  4. The $L^\infty$ estimates
  5. Estimates of $\|n+\Delta\varphi\|_{s}$
  6. Estimates of $\|\nabla F\|$
  7. Estimates of $\|\nabla\varphi\|_\infty$
  8. Estimates of $\|n+\Delta_g\varphi\|_\infty$
  9. Proof of Theorem \ref{['theo:1.1']}
  10. The Hölder estimates for parabolic equations
  11. An elementary lemma

Key Result

Theorem 1.1

Let $(M, \omega_g)$ be a compact Kähler manifold of complex dimension $n\geq 2$, and $\{\varphi(t), t\in [0, T)\}$ the solution to the Calabi flow (eq000) with $T<\infty.$ If the scalar curvature satisfies for $p>n$, the Calabi flow can be extended past time $T$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 29 more