Parabolic Frequency and Its Applications Under the Geometric Flow

This paper investigates the parabolic frequency of solutions to the linear heat equations under the geometric flow. In the first part, we prove frequency monotonicity of a solution of the heat equation with a spacetime potential. This provides a unifying approach that not only recovers the related results on the Ricci flow, but also yields new applications, particularly concerning backward uniqueness and eigenvalue estimates, for the Yamabe flow. Second, for the case of a time‐dependent potential, we employ an elliptic‐type gradient estimate to establish frequency monotonicity under a significantly weakened curvature assumption. From which, we derive an integral‐type Harnack inequality.