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The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds

Shuang Liang, Xi Sisi Shen, Kevin Smith

Abstract

We prove local Calabi and higher order estimates for solutions to the continuity equation introduced by La Nave-Tian and extended to Hermitian metrics by Sherman-Weinkove. We apply the estimates to show that on a compact complex manifold the Chern scalar curvature of a solution must blow up at a finite-time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus-Toma manifolds we prove Gromov-Hausdorff and smooth convergence of the metric to a particular non-negative $(1,1)$-form as $t\to\infty$.

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds

Abstract

We prove local Calabi and higher order estimates for solutions to the continuity equation introduced by La Nave-Tian and extended to Hermitian metrics by Sherman-Weinkove. We apply the estimates to show that on a compact complex manifold the Chern scalar curvature of a solution must blow up at a finite-time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus-Toma manifolds we prove Gromov-Hausdorff and smooth convergence of the metric to a particular non-negative -form as .
Paper Structure (11 sections, 9 theorems, 114 equations)

This paper contains 11 sections, 9 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local Calabi estimates
  4. $C^3$ estimate
  5. Higher order estimates
  6. Blow-up of the scalar curvature at a finite time singularity
  7. Oeljeklaus-Toma manifolds
  8. $C^0$ estimate
  9. $C^2$ estimate
  10. Higher order estimates
  11. Ricci curvature bounds

Key Result

Theorem 1

Let $\chi(t)$ solve the continuity equation Intro:ContinuityEquation or the normalized continuity equation Intro:NormalizedContinuityEquation starting at a Hermitian metric $\chi_0$ in a neighborhood of $B_r$ for fixed $r\in (0,1)$ and $t\in [0,T]$. Assume that there exists a Hermitian metric $\hat{ Then for any fixed $\varepsilon > 0$ there exists a uniform constant $C$ depending only on $K,\chi_

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • proof
  • proof
  • proof : Proof of Theorem \ref{['Intro:ThmBlowup']}
  • Proposition 1
  • proof
  • ...and 9 more