Higher-order Ricci estimates along immortal K\"ahler-Ricci flows

TL;DR

This paper investigates higher-order curvature estimates along K"ahler-Ricci flows on compact K"ahler manifolds with intermediate Kodaira dimension, identifying conditions for uniform bounds and obstructions to higher-order regularity.

Contribution

It establishes uniform curvature bounds away from singular fibers and identifies a geometric obstruction to higher-order bounds in K"ahler-Ricci flows.

Findings

Ricci curvature is uniformly bounded in $C^1$ away from singular fibers

A geometric obstruction causes third-order Ricci derivatives to blow up at rate $e^{t/2}$

Uniform $C^k$ bounds hold in isotrivial and torus-fibered cases

Abstract

We study higher-order curvature estimates along K\"ahler-Ricci flows on compact K\"ahler 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.