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Long-time existence of the Laplacian flow with Weyl tensor
Abstract
We establish integral extension criteria for the Laplacian flow of closed G2-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 Lp norm of the $77$-dimensional Weyl component W77 remains uniformly bounded for some p≥ 7/2. We also obtain a mixed space--time extension criterion under the condition W77∈ Lq(0,T;Lp(M,g(t))), 7/2<p<∞, 1≤ q<∞, 1/q+7/p≤ 2. In particular, finite space-time L4 integrability of W77 is sufficient for extension.
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BibTeX
@misc{li2026longtime,
title = {{Long-time existence of the Laplacian flow with Weyl tensor}},
author = {Chuanhuan Li and Yi Li and Kai Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23412},
eprint = {2610.23412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23412},
}
RIS
TY - UNPB
TI - Long-time existence of the Laplacian flow with Weyl tensor
AU - Chuanhuan Li
AU - Yi Li
AU - Kai Zheng
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23412
UR - https://arxiv.org/abs/2610.23412
AB - 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^p norm of the $77$-dimensional Weyl component W_77 remains uniformly bounded for some p≥ 7/2. We also obtain a mixed space--time extension criterion under the condition W_77∈ L^q(0,T;L^p(M,g(t))), 7/2<p<∞, 1≤ q<∞, 1/q+7/p≤ 2. In particular, finite space-time L^4 integrability of W_77 is sufficient for extension.
ER -