Gradient estimates and Parabolic frequency under the Laplacian $G_{2}$ flow

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发表日期：

2025年3月24日

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Chuanhuan Li,
Yi Li,
Kairui Xu

摘要

In this paper, we consider the Laplacian $G_{2}$ $G_{2}$ flow on a closed seven-dimensional manifold M with a closed $G_{2}$ $G_{2}$ -structure. We first obtain the gradient estimates for positive solutions of the heat equation under the Laplacian $G_{2}$ $G_{2}$ flow and then we get the Harnack inequality on spacetime. As an application, we prove the monotonicity of parabolic frequency 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 solutions of the linear heat equation with bounded Bakry-Émery Ricci curvature, and then obtain the backward uniqueness.

Gradient estimates and Parabolic frequency under the Laplacian \(G_{2}\) flow

摘要