Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds

Abstract

We prove Cohn--Vossen-type scalar curvature inequalities on complete noncompact Riemannian manifolds with nonnegative Ricci curvature, motivated by Yau's higher-dimensional problem. For $n\ge 3$, we obtain an $O(r^{n-2})$ normalized growth estimate under a ${\mathbb{Z}}^{n-2}$ subgroup condition on ${\pi }_1(M)$. For locally conformally flat manifolds, we prove the corresponding normalized estimate in the non-${\mathbb{R}}^n$ case and derive polynomial or exponential upper bounds in the conformally Euclidean case. In dimension three, we prove the sharp asymptotic scalar-curvature flux estimate $8\pi$ under quadratic scalar-curvature decay, confirming the Munteanu--Wang conjectural $8\pi$ bound in this setting; we also prove finiteness of the flux for manifolds with a foliated end. Finally, under the Cohn--Vossen-scale scalar-growth hypothesis, we prove weighted analogues for the weighted scalar curvature ${\mathrm{Sc}}_{\alpha ,\beta }$ on weighted Riemannian manifolds with nonnegative Bakry--Émery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry--Émery regimes.

This version was submitted to a journal in June 2026