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Total scalar curvature under a curvature operator lower bound
Abstract
Let (Mn, g) be a complete, simply connected Riemannian manifold without boundary, of dimension n≥3, with curvature operator at least that of the unit sphere. We prove that ∫M scal(x) d Volx≤ n(n−1)ωn, where ωn is the volume of the unit n-sphere. Equality holds if and only if (M,g) is isometric to the unit round sphere. In fact, we obtain a stronger bound containing Vol(M, g). In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.
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BibTeX
@misc{ge2026total,
title = {{Total scalar curvature under a curvature operator lower bound}},
author = {Jian Ge and Chuanhuan Li and Ronggang Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19851},
eprint = {2609.19851},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19851},
}
RIS
TY - UNPB
TI - Total scalar curvature under a curvature operator lower bound
AU - Jian Ge
AU - Chuanhuan Li
AU - Ronggang Li
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19851
UR - https://arxiv.org/abs/2609.19851
AB - Let (M^n, g) be a complete, simply connected Riemannian manifold without boundary, of dimension n≥3, with curvature operator at least that of the unit sphere. We prove that ∫_M scal(x) d Vol_x≤ n(n−1)ω_n, where ω_n is the volume of the unit n-sphere. Equality holds if and only if (M,g) is isometric to the unit round sphere. In fact, we obtain a stronger bound containing Vol(M, g). In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.
ER -