闭G_2结构的Laplacian流：Shi型估计、唯一性和compactness
Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
摘要：
我们为封闭G_2结构的Laplacian流发展了基础理论，这将是未来研究的基础。（一）.我们证明了Riemann曲率张量Rm和挠率张量T沿流的Shi型导数估计，即$Λ(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$上的一个界意味着Rm和T的所有共变导数上的界.（二）、我们证明了$Λ(x,t)$将在有限时间奇点处爆破，因此只要$Λ(x,t)$保持有界，流动就将存在。（三）、给出了流的前向唯一性和后向唯一性的新证明，并给出了一些应用。（四）、我们证明了流的一个紧性定理，并利用它加强了（2）的长期存在性结果。（五）、最后，我们研究了Laplacian流的紧孤子解。
英文摘要：
We develop foundational theory for the Laplacian flow for closed G_2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on $Λ(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $Λ(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $Λ(x,t)$ remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2). (5). Finally, we study compact soliton solutions of the Laplacian flow.
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