Manifolds for which Huber's Theorem holds

Extensions of Huber's Theorem to higher dimensions with $L^\frac{n}{2}$ bounded scalar curvature have been extensively studied over the years. In this paper, we delve into the properties of conformal metrics on a punctured ball with $\|R\|_{L^\frac{n}{2}}<+\infty$, aiming to identify necessary geometric constraints for Huber's theorem to be applicable. Unexpectedly, such metrics are more rigid than we initially anticipated. For instance, we found that the volume density at infinity is precisely one, and the blow-down of the metric is $\mathbb{R}^n$. Specifically, in four dimensions, we derive the $L^2$-integrability of the Ricci curvature, which directly leads to the conclusion that the Pfaffian 4-form is integrable and adheres to a Gauss-Bonnet-Chern formula. Additionally, we demonstrate that a Gauss-Bonnet-Chern formula, previously verified by Lu and Wang under the assumption that the second fundamental form belongs to $L^4$, remains valid for $R \in L^2$. Consequently, on an orientable 4-dimensional manifold conformal to a domain in a closed manifold, Huber's Theorem holds when $R \in L^2$, if and only if the negative part of the Pfaffian 4-form is integrable.