Parabolic frequency monotonicity on compact manifolds

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The proof is based on a generalization of R. Hamilton's matrix Harnack inequality for small time. As applications, we obtain a unique continuation result. Monotonicity of a new quantity under two-dimensional Ricci flow, closely related to the parabolic frequency functional, is derived as well.