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Higher-order Ricci estimates along immortal Kähler-Ricci flows

Wenrui Kong

Abstract

We study higher-order curvature estimates along Kähler-Ricci flows on compact Kähler manifolds of intermediate Kodaira dimension. We prove that away from singular fibers, the Ricci curvature is uniformly bounded in $C^1$, the Laplacian of the Ricci curvature in $C^0$, and the scalar curvature in $C^2$. We identify a geometric obstruction to higher-order curvature bounds, whose non-vanishing causes a specific third-order derivative of the Ricci curvature to blow up at rate $e^{t/2}$. Uniform $C^k$ bounds for every $k$ hold for the Ricci curvature in the isotrivial case, and for the full Riemann curvature in the torus-fibered case.

Higher-order Ricci estimates along immortal Kähler-Ricci flows

Abstract

We study higher-order curvature estimates along Kähler-Ricci flows on compact Kähler manifolds of intermediate Kodaira dimension. We prove that away from singular fibers, the Ricci curvature is uniformly bounded in , the Laplacian of the Ricci curvature in , and the scalar curvature in . We identify a geometric obstruction to higher-order curvature bounds, whose non-vanishing causes a specific third-order derivative of the Ricci curvature to blow up at rate . Uniform bounds for every hold for the Ricci curvature in the isotrivial case, and for the full Riemann curvature in the torus-fibered case.

Paper Structure

This paper contains 20 sections, 54 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Setup and Main Results
  3. Paper Outline
  4. Acknowledgements
  5. Preliminaries
  6. Known Estimates
  7. Parabolic Hölder Norm
  8. Asymptotic Expansion
  9. First-Order Ricci Estimates
  10. Proof of Theorem 1.4
  11. Proof of Theorem 3.1
  12. Second-Order Curvature Estimates
  13. Proof of Theorem 1.5
  14. Proof of Theorem 1.2
  15. Hessian of Ricci and Proof of Theorem 1.3
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(X,\omega_0)$ be a compact Kähler manifold with $K_X$ semiample and intermediate Kodaira dimension $0 < m < \dim X$, and let $\omega^\bullet(t)$ solve eq_krf. Given any $K \Subset X \setminus S$, there exists a constant $C_{K}$ such that

Theorems & Definitions (115)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 105 more