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In this note we obtain local derivative estimates of Shi-type for the heatequation coupled to the Ricci flow. As applications, in part combining withKuang's work, we extend some results of Zhang and Bamler-Zhang includingdistance distortion estimates and a backward pseudolocality theorem for Ricciflow on compact manifolds to the noncompact case.

Local derivative estimates for the heat equation coupled to the Ricci flow