On the structure of complete $G_2$-solitons

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 $C^{1,\alpha }$ convergence. Our principal result shows that this $C^{1,\alpha }$ 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 $C^{1,\alpha }$ 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 $\epsilon$-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed $C^{1,\alpha }$ 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 $L^{\frac{7}{2}}$ -curvature bound then yields pointed $C^{\infty }$ compactness.