Small Normal Curvature and Three-Manifold Topology
Abstract
For m=2,3, we prove that every smooth immersion F:ℝℙmlooparrowright𝔹̅,N(1) satisfies κ(F)2≥ 2m/(m+1), with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with κ(F)≤√(3/2) is diffeomorphic to S3, ℝℙ3, or S2× S1. All three possibilities occur, while κ(F)<√(3/2) forces X≅ S3. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality (minY Rg)sys(g)2<6π2 for every spherical three-space form Y with |π1(Y)|>2. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for ℝℙ3 of Bray--Brendle--Eichmair--Neves.
Related
- 4-manifold topology and collision rank of groups
- Convexity of a ratio function involving polygamma functions and monotonicity of the sectional curvature of the Fisher--Rao manifold
- Uniformly Positive Scalar Curvature on Manifold Interiors
- Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
- Counterexamples to preservation of flat normal bundles under mean curvature flow
BibTeX
@misc{chow2026small,
title = {{Small Normal Curvature and Three-Manifold Topology}},
author = {Tsz-Kiu Aaron Chow and Jingbo Wan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18002},
eprint = {2608.18002},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18002},
}
RIS
TY - UNPB
TI - Small Normal Curvature and Three-Manifold Topology
AU - Tsz-Kiu Aaron Chow
AU - Jingbo Wan
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.18002
UR - https://arxiv.org/abs/2608.18002
AB - For m=2,3, we prove that every smooth immersion F:ℝℙ^mlooparrowright𝔹̅^,N(1) satisfies κ(F)^2≥ 2m/(m+1), with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with κ(F)≤√(3/2) is diffeomorphic to S^3, ℝℙ^3, or S^2× S^1. All three possibilities occur, while κ(F)<√(3/2) forces X≅ S^3. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality (min_Y R_g)sys(g)^2<6π^2 for every spherical three-space form Y with |π_1(Y)|>2. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for ℝℙ^3 of Bray--Brendle--Eichmair--Neves.
ER -