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تسجيل مستخدم جديدManifolds for which Hubers Theorem holds
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Extensions of Hubers finite-point conformal compactification theorem to higher dimensions with $L^frac{n}{2}$ bounded scalar curvature have been studied for many years. In this paper, we discuss the properties of conformal metrics with $|R|_{L^frac{n}{2}}<+infty$ on a punctured ball of a Riemannian manifold to find some geometric obstacles for Hubers theorem. To our surprise, such metrics are rather more rigid than we have ever thought. For example, their volume densities at infinity are exact 1, which implies that Carron and Herzlichs Euclidean volume growth condition is also a necessary condition for Hubers Theorem. When the dimension is 4, we derive the $L^2$-integrability of Ricci curvature, which follows that the Pfaffian of the curvature is integrable and satisfies a Gauss-Bonnet-Chern formula. We also prove that the Gauss-Bonnet-Chern formula proved by Lu and Wang, under the assumption that the second fundamental form is in $L^4$, holds when $Rin L^2$.
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