On the structure of complete G2-solitons
Abstract
In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed G2-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed C1,α convergence. Our principal result shows that this C1,α convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive 3-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to G2-solitons. Since the available C1,α control does not directly close an elliptic bootstrap for the G2-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy ε-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed C1,α convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with G2-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted L7/2 -curvature bound then yields pointed C∞ compactness.
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BibTeX
@misc{li2026structure,
title = {{On the structure of complete $G_2$-solitons}},
author = {Haozhao Li and Yuanqing Ma and Kai Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.06619},
eprint = {2606.06619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06619},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the structure of complete G_2-solitons
AU - Haozhao Li
AU - Yuanqing Ma
AU - Kai Zheng
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2606.06619
UR - https://arxiv.org/abs/2606.06619
AB - In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed G_2-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed C^1,α convergence. Our principal result shows that this C^1,α convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive 3-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to G_2-solitons. Since the available C^1,α control does not directly close an elliptic bootstrap for the G_2-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy ε-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed C^1,α convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with G_2-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted L^7/2 -curvature bound then yields pointed C^∞ compactness.
ER -