In this paper we give local curvature estimates for the Laplacian flow on closed \(G_{2}\)-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed \(G_{2}\)-structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) result which is an analogue of Sesum’s theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed \(G_{2}\)-structures. Roughly speaking, we can prove that the time derivative of the scalar curvature \(R_{g(t)}\) is equal to the Laplacian of \(R_{g(t)}\), plus an extra term which can be written as the difference of two nonnegative quantities.

在本文中，我们给出了在Ricci曲率沿流约束的条件下，封闭\（G_ {2} \）结构上Laplacian流的局部曲率估计。主要成分包括Kotschwar等人的想法。（J Funct Anal 271（9）：2604–2630，2016），他给出了完整流形上Ricci流的局部曲率估计，然后提供了Sesum结果的新基本证明（Am J Math 127（6）：1315– 1324，2005），以及封闭\（G_ {2} \）上拉普拉斯流的特殊结构结构。立刻产生的结果是，该估计值给出了Lotay和Wei（Geom Funct Anal 27（1）：165–233，2017）结果的新证明，该结果与Sesum定理类似。第二个结果是关于一个封闭的\（G_ {2} \）结构的Laplacian流的标量曲率的有趣的演化方程。粗略地说，我们可以证明数量曲率的时间导数\（R_ {克（T）} \）等于的拉普拉斯\（R_ {克（T）} \） ，再加上一个额外的术语，其可以是表示为两个非负量的差。