PreprintCC BY 4.0arXiv
Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity
Abstract
For any complete noncompact manifolds (Mn, g) of nonnegative sectional curvature, with 3≤ n≤6, we obtain limr→∞r2−n∫Bp(r)Scg dVg≤8πωn−2(1−AVR(Mn,g)) and the corresponding rigidity. The latter bound and its equality classification hold in (Mn, g) with positive-dimensional souls. With a pole, limr→∞r2−n∫Bp(r)Scg dVg≤4πωn−2(1−AVR(Mn,g)) follows from an independent distance-sphere proof. We also prove the sharp total scalar-integral bound for closed manifolds whose c
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BibTeX
@misc{xu2026integral,
title = {{Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity}},
author = {Guoyi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26357},
eprint = {2609.26357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26357},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity
AU - Guoyi Xu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26357
UR - https://arxiv.org/abs/2609.26357
AB - For any complete noncompact manifolds (M^n, g) of nonnegative sectional curvature, with 3≤ n≤6, we obtain lim_r→∞r^2−n∫_B_p(r)Sc_g dV_g≤8πω_n−2(1−AVR(M^n,g)) and the corresponding rigidity. The latter bound and its equality classification hold in (M^n, g) with positive-dimensional souls. With a pole, lim_r→∞r^2−n∫_B_p(r)Sc_g dV_g≤4πω_n−2(1−AVR(M^n,g)) follows from an independent distance-sphere proof. We also prove the sharp total scalar-integral bound for closed manifolds whose c
ER -