Gradient estimates and parabolic frequency under the Laplacian G_2 flow

TL;DR

This paper studies the Laplacian G_2 flow on 7-manifolds, deriving gradient estimates, Harnack inequalities, and monotonicity properties for heat equation solutions, with implications for backward uniqueness.

Contribution

It provides new gradient estimates and monotonicity results for heat equations under the Laplacian G_2 flow, extending understanding of geometric flows on G_2-structures.

Findings

Established gradient estimates for heat solutions under the flow.

Proved Harnack inequalities and their spacetime versions.

Demonstrated monotonicity of parabolic frequency and backward uniqueness.

Abstract

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.