Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds
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| Format: | Preprint |
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2026
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| _version_ | 1866911738556841984 |
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| author | Deng, Jialong |
| author_facet | Deng, Jialong |
| contents | We prove Cohn-Vossen-type scalar-curvature inequalities on complete noncompact Riemannian manifolds with nonnegative Ricci curvature, motivated by Yau's higher-dimensional problem. In dimensions n >= 3, we obtain a normalized O(r^{n-2}) growth estimate under the assumption that the fundamental group contains a free abelian subgroup of rank n-2. For locally conformally flat manifolds, we prove the corresponding normalized estimate outside the topological Euclidean case and derive polynomial or exponential upper bounds in the conformally Euclidean case.
In dimension three, under quadratic scalar-curvature decay, we prove the sharp asymptotic scalar-curvature flux upper bound 8 pi (1 - AVR(g)). This confirms the Munteanu-Wang conjectural 8 pi bound in this setting and refines it by an asymptotic-volume-ratio correction. 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 on weighted Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry-Emery regimes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01368 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds Deng, Jialong Differential Geometry We prove Cohn-Vossen-type scalar-curvature inequalities on complete noncompact Riemannian manifolds with nonnegative Ricci curvature, motivated by Yau's higher-dimensional problem. In dimensions n >= 3, we obtain a normalized O(r^{n-2}) growth estimate under the assumption that the fundamental group contains a free abelian subgroup of rank n-2. For locally conformally flat manifolds, we prove the corresponding normalized estimate outside the topological Euclidean case and derive polynomial or exponential upper bounds in the conformally Euclidean case. In dimension three, under quadratic scalar-curvature decay, we prove the sharp asymptotic scalar-curvature flux upper bound 8 pi (1 - AVR(g)). This confirms the Munteanu-Wang conjectural 8 pi bound in this setting and refines it by an asymptotic-volume-ratio correction. 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 on weighted Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry-Emery regimes. |
| title | Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2606.01368 |