Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

Haoyuan Sun

Abstract

We establish uniform diameter estimates and volume non-collapsing estimates for the Chern-Ricci flow on smooth Hermitian minimal models of general type, assuming the initial metric is Kähler in a neighborhood of the null locus of the canonical bundle. This yields subsequential Gromov-Hausdorff convergence, partially resolving a conjecture of Tosatti and Weinkove. When the underlying manifold is Kähler, we further prove the uniqueness of the limit space. Analytically, we overcome the difficulties posed by non-Kähler torsion in the Green's formula by exploiting our local Kähler assumption, successfully adapting recent estimates of Kähler Green's function to the Hermitian setting. To prove the uniqueness of the limit, we introduce Perelman's reduced length to the Chern-Ricci flow. By establishing a uniform Chern scalar curvature bound and an almost monotonicity formula for the reduced volume, we deduce an almost-avoidance principle for the singular set, allowing us to effectively compare the flow distance with the canonical limit distance.

Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

Abstract

We establish uniform diameter estimates and volume non-collapsing estimates for the Chern-Ricci flow on smooth Hermitian minimal models of general type, assuming the initial metric is Kähler in a neighborhood of the null locus of the canonical bundle. This yields subsequential Gromov-Hausdorff convergence, partially resolving a conjecture of Tosatti and Weinkove. When the underlying manifold is Kähler, we further prove the uniqueness of the limit space. Analytically, we overcome the difficulties posed by non-Kähler torsion in the Green's formula by exploiting our local Kähler assumption, successfully adapting recent estimates of Kähler Green's function to the Hermitian setting. To prove the uniqueness of the limit, we introduce Perelman's reduced length to the Chern-Ricci flow. By establishing a uniform Chern scalar curvature bound and an almost monotonicity formula for the reduced volume, we deduce an almost-avoidance principle for the singular set, allowing us to effectively compare the flow distance with the canonical limit distance.

Paper Structure

This paper contains 15 sections, 50 theorems, 358 equations.

Table of Contents

  1. Introduction
  2. Preliminary estimates for the Chern-Ricci flow
  3. Elliptic tools
  4. Estimates for families of Laplacian equations
  5. Estimates of Green's functions
  6. Diameter and volume non-collapsing estimates
  7. Bounding the Chern scalar curvature along the flow
  8. Perelman's reduced length and reduced volume
  9. The $\mathcal{L}$-Geodesic equation and the $\mathcal{L}$-exponential map
  10. Derivatives of the reduced length
  11. The Laplacian bound of $L$
  12. Almost monotonicity of the reduced volume
  13. Reduction to a key estimate
  14. An almost avoidance principle
  15. End of proof of the key estimate

Key Result

Theorem 1.1

In the setting of conjecture, if $\omega_0$ is Kähler in an arbitrary neighborhood of $E=\operatorname{Null}(K_X)$, then there are uniform constants $c,C,\alpha,r_0$ such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (103)

  • Conjecture 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.1
  • ...and 93 more