Mean Value Inequalities and Conditions to Extend Ricci Flow

This paper concerns conditions related to the first finite singularity time of a Ricci flow solution on a closed manifold. In particular, we provide a systematic approach to the mean value inequality method, suggested by N. Le and F. He. We also display a close connection between this method and time slice analysis of B. Wang. As an application, we prove several inequalities for a Ricci flow solution on a manifold with nonnegative isotropic curvature.