PreprintCC BY 4.0arXiv
Integral of Scalar Curvature under Volume Collapse
Abstract
We construct a smooth complete metric on R3 with strictly positive Ricci curvature and lims→∞s−1∫Bp(s)Scddμ= ∞. This gives a negative answer to Shing-Tung Yau's question on the asymptotic scalar-curvature integral. We also give counterexamples to Naber's local Shing-Tung Yau conjecture on compact manifolds. The noncompact example has zero asymptotic volume ratio, and the compact counterexamples have volumes tending to zero.
Related
- A Counterexample to Yau's Conjectured Asymptotic Scalar-Curvature Integral Bound
- Integral of Scalar Curvature: Sharp Asymptotic Bounds and Rigidity
- Scalar curvature, volume, and Weil-Petersson translation length
- Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature
- Volume growth and integral curvature bound for non-negatively curved three-manifolds
BibTeX
@misc{xu2026integral,
title = {{Integral of Scalar Curvature under Volume Collapse}},
author = {Guoyi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10160},
eprint = {2609.10160},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10160},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integral of Scalar Curvature under Volume Collapse
AU - Guoyi Xu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10160
UR - https://arxiv.org/abs/2609.10160
AB - We construct a smooth complete metric on R^3 with strictly positive Ricci curvature and lim_s→∞s^−1∫_B_p(s)Scddμ= ∞. This gives a negative answer to Shing-Tung Yau's question on the asymptotic scalar-curvature integral. We also give counterexamples to Naber's local Shing-Tung Yau conjecture on compact manifolds. The noncompact example has zero asymptotic volume ratio, and the compact counterexamples have volumes tending to zero.
ER -