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≥3, we obtain an O(rn−2) normalized growth estimate under a ℤn−2 subgroup condition on π1(M). For locally conformally flat manifolds, we prove the corresponding normalized estimate in the non-ℝn 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 8π under quadratic scalar-curvature decay, confirming the Munteanu--Wang conjectural 8π 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 Scα,β on weighted Riemannian manifolds with nonnegative Bakry--Émery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry--Émery regimes.
Related
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BibTeX
@misc{deng2026cohnvossentype,
title = {{Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds}},
author = {Jialong Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.01368},
eprint = {2606.01368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01368},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds
AU - Jialong Deng
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2606.01368
UR - https://arxiv.org/abs/2606.01368
AB - 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≥3, we obtain an O(r^n−2) normalized growth estimate under a ℤ^n−2 subgroup condition on π_1(M). For locally conformally flat manifolds, we prove the corresponding normalized estimate in the non-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 8π under quadratic scalar-curvature decay, confirming the Munteanu--Wang conjectural 8π 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 Sc_α,β on weighted Riemannian manifolds with nonnegative Bakry--Émery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry--Émery regimes.
ER -