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Volume growth and integral curvature bound for non-negatively curved three-manifolds

  • Pak-Yeung Chan
  • Man-Chun Lee
  • Mingxiang Li
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Abstract

Motivated by results in Kähler geometry, in this work, we are interested in understanding the relation between integral curvature bounds and volume growth, under non-negative curvature in dimension three. In case of non-negative sectional curvature, we show that for metric on Euclidean space, it is of Euclidean volume growth if and only if it has average quadratic curvature decay. This is based on showing that metrics on three-dimensional Euclidean space with non-negative sectional curvature is of Euclidean volume growth if its asymptotic scaling invariant integral of scalar curvature is smaller than the sharp constant 8π. We also show a gap Theorem if the curvature decay fast enough in the average sense, under non-negative Ricci curvature.

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Cite

BibTeXRIS
BibTeX
@misc{chan2026volume,
  title = {{Volume growth and integral curvature bound for non-negatively curved three-manifolds}},
  author = {Pak-Yeung Chan and Man-Chun Lee and Mingxiang Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24096},
  eprint = {2609.24096},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24096},
}
RIS
TY  - UNPB
TI  - Volume growth and integral curvature bound for non-negatively curved three-manifolds
AU  - Pak-Yeung Chan
AU  - Man-Chun Lee
AU  - Mingxiang Li
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24096
UR  - https://arxiv.org/abs/2609.24096
AB  - Motivated by results in Kähler geometry, in this work, we are interested in understanding the relation between integral curvature bounds and volume growth, under non-negative curvature in dimension three. In case of non-negative sectional curvature, we show that for metric on Euclidean space, it is of Euclidean volume growth if and only if it has average quadratic curvature decay. This is based on showing that metrics on three-dimensional Euclidean space with non-negative sectional curvature is of Euclidean volume growth if its asymptotic scaling invariant integral of scalar curvature is smaller than the sharp constant 8π. We also show a gap Theorem if the curvature decay fast enough in the average sense, under non-negative Ricci curvature.
ER  -