A local curvature estimate for the Ricci flow

TL;DR

This paper provides a local curvature estimate for Ricci flow solutions, offering a new proof of bounded curvature under Ricci bounds and analyzing singularity formation.

Contribution

It introduces an explicit local curvature estimate and extends known results to noncompact manifolds, also analyzing singularity blow-up rates.

Findings

Curvature can be explicitly estimated locally in terms of initial data and Ricci bounds.

On compact manifolds, curvature cannot blow up if Ricci remains bounded.

Ricci curvature must blow up at least linearly at finite time singularities.

Abstract

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result of Sesum, which asserts that the curvature of a solution on a compact manifold cannot blow up while the Ricci curvature remains bounded, and extends its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.