I will talk about the 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 Lei Ni. 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. This is a joint work with Xiaolong Li (UC Irivine).
