Scalar Curvature Lower Bounds and Convergence in Measure
Abstract
Assume that smooth metrics gk converge in measure to a smooth metric g. Gromov asked the following question: if all the metrics gk have non-negative scalar curvature does it follow that g also has non-negative scalar curvature? We answer Gromov's question in dimension three. We show that, without further assumptions, the metric g need not have non-negative scalar curvature and in fact g may be completely arbitrary. However, if one assumes in addition that the identity maps (M,gk)→ (M,g) are uniformly bi-Lipschitz, then indeed g must have non-negative scalar curvature. Our method of proof can also be used to show that, under an almost Euclidean entropy condition, scalar curvature lower bounds will persist under convergence of the metrics together with their inverses in Lp for suitably large p.
Related
- An improved volume bound under Ricci and scalar curvature lower bounds
- Macroscopic scalar curvature bounds on surfaces
- Total scalar curvature under a curvature operator lower bound
- Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class
- Scalar curvature bounds and rigidity of Ricci shrinkers
BibTeX
@misc{mazurowski2026scalar,
title = {{Scalar Curvature Lower Bounds and Convergence in Measure}},
author = {Liam Mazurowski and Xuan Yao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34141},
eprint = {2609.34141},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34141},
}
RIS
TY - UNPB
TI - Scalar Curvature Lower Bounds and Convergence in Measure
AU - Liam Mazurowski
AU - Xuan Yao
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34141
UR - https://arxiv.org/abs/2609.34141
AB - Assume that smooth metrics g_k converge in measure to a smooth metric g. Gromov asked the following question: if all the metrics g_k have non-negative scalar curvature does it follow that g also has non-negative scalar curvature? We answer Gromov's question in dimension three. We show that, without further assumptions, the metric g need not have non-negative scalar curvature and in fact g may be completely arbitrary. However, if one assumes in addition that the identity maps (M,g_k)→ (M,g) are uniformly bi-Lipschitz, then indeed g must have non-negative scalar curvature. Our method of proof can also be used to show that, under an almost Euclidean entropy condition, scalar curvature lower bounds will persist under convergence of the metrics together with their inverses in L^p for suitably large p.
ER -