Gradient estimates and parabolic frequency under the Laplacian G_2 flow

In this paper, we consider the Laplacian G_2 flow on a closed seven-dimensional manifold M with a closed G_2-structure. We first obtain the gradient estimates of positive solutions of the heat equation under the Laplacian G_2 flow and then we get the Harnack inequality on spacetime. As an application, we prove the monotonicity for positive solutions of the heat equation with bounded Ricci curvature, and get the integral-type Harnack inequality. Besides, we prove the monotonicity of parabolic frequency for positive solutions of the linear heat equation with bounded Bakry-Emery Ricci curvature, and then obtain the backward uniqueness.