Three Circles Theorem for Volume of Conformal Metrics

In this paper, we establish three circles theorem for volume of conformal metrics whose scalar curvatures are integrable in a critical (scaling invariant) norm. As applications, we analyze the asymptotic behavior of such metrics near isolated singularities and use it to show the residual terms of the Chern–Gauss–Bonnet formula are integers. Such strong rigidity implies a vanishing theorem on the integral value of the \(Q_g\) $Q_g$ curvature, with application to the bi-Lipschitz equivalence problem for conformal metrics.