Extension under bounded torsion or scalar curvature

Determine whether the Laplacian flow on closed $G_2$-structures on compact 7-dimensional manifolds exists for all finite times whenever the torsion tensor or scalar curvature remains uniformly bounded.

Background

The paper studies finite-time extension criteria for the Laplacian flow of closed G2G_2-structures. Its main results establish smooth extension under a uniform lower bound for scalar curvature together with spatial or mixed space-time integral bounds on the W77W_{77} component of the Weyl tensor.

The cited question is broader: it asks whether uniform boundedness of the torsion tensor or scalar curvature alone prevents finite-time singularities. The results in the paper do not resolve this general question; they instead provide extension criteria involving integral control of W77W_{77} in addition to a scalar-curvature lower bound.

References

Lotay and Wei propose the following question (see also ): \begin{quote} {\bf Question:} {\it whether the Laplacian $G_{2}$ flow on closed $G_{2}$-structures will exist as long as the torsion tensor or scalar curvature remains uniformly bounded.} \end{quote}

— Long-time existence of the Laplacian flow with Weyl tensor  (2610.23412 - Li et al., 16 Oct 2026) in Section 1, Introduction