Integral curvature estimates for solutions to Ricci flow with $L^p$ bounded scalar curvature

TL;DR

The paper develops localised weighted curvature integral estimates for Ricci flow in the 4D or closed Kähler-Ricci flow setting, providing a framework to bound curvature through weighted $L^2$-type integrals. The method introduces a time-dependent weight and derives a basic inequality that controls Ricci components; in 4D or Kahler contexts, full curvature terms are controlled via Gauss-Bonnet or Kaehler integral identities, linking $|{ m Rm}|^2$ to $|{ m Ric}|^2$ and $R^2$. Under a spatial $L^p$ bound on the scalar curvature with $p>2$, these estimates yield a uniform $L^2$-bound on the Riemann curvature, with stronger bounds under a weak non-inflating condition or when the manifold is closed, with boundary contributions carefully managed. The results contribute to local curvature control and pave the way toward non-collapsing and higher-regularity conclusions for short-time evolution, drawing on detailed curvature identities and boundary-term analysis.

Abstract

In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial $L^p$ sense for some $p>2,$ then the estimates imply a uniform bound on the spatial $L^2$ norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4