Heat Kernel Estimate Along Ricci-Harmonic Flow

In this paper, we study the Ricci-harmonic flow under the assumption that the scalar curvature is bounded. First, we establish a time-derivative bound for solutions to the heat equation along the flow. Based on this estimate, we derive a short-time distance-distortion estimate and prove the existence of suitable cutoff functions. Using these results, we obtain Gaussian-type upper and lower bounds for the heat kernel along the Ricci-harmonic flow. Our results generalize the previous work of Bamler–Zhang on Ricci flow to the Ricci-harmonic flow setting, and can be used to study the regularity theory of Ricci-harmonic flow.