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

TL;DR

This paper establishes localized weighted curvature integral estimates for Ricci flow solutions, demonstrating that bounded scalar curvature in an $L^p$ sense leads to uniform bounds on the curvature tensor, extending previous results.

Contribution

It introduces new integral curvature estimates for Ricci flow under scalar curvature bounds, improving upon earlier work and covering more general settings.

Findings

Bounded scalar curvature in $L^p$ implies uniform $L^2$ bounds on curvature tensor.

Stronger estimates are obtained under non-inflating conditions or on closed manifolds.

The results extend previous integral curvature estimates to broader contexts.

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 K\"ahler 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.