Long-time existence of the Laplacian flow with Weyl tensor

Abstract: We establish integral extension criteria for the Laplacian flow of closed $G_2$-structures on compact $7$-dimensional manifolds $M$. Assuming a uniform lower bound for the scalar curvature, we prove that a finite-time solution extends smoothly whenever the spatial $L<sup>p$ norm of the $77$-dimensional Weyl component $W_{77}$ remains uniformly bounded for some $p\ge 7/2$. We also obtain a mixed space--time extension criterion under the condition [ W_{77}\in L^q(0,T;L^p(M,g(t))), \qquad \frac{7}{2}<p<\infty,\quad 1\le q<\infty,\quad \frac{1}{q}+\frac{7}{p}\le 2. ] In particular, finite space-time $L<sup>4$ integrability of $W_{77}$ is sufficient for extension.