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A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive σk-curvature Metrics
Abstract
Let k∈ℕ satisfy 1<k<n/2, and let Σp⊂Snn be a closed smooth embedded submanifold. We prove that a complete conformal metric g=v−2gSnn on Snn∖Σ satisfying λ(g−1Ag)∈Gk and σk(g−1Ag)=κ>0 must obey p≤ pk(n), where pk is the model threshold determined by Hhp+1×Snn−p−1. When k=2 and n=m2, we construct a smooth complete equality example on Snn∖Sn(m2−m−2)/2. We also prove that the strict inequality p<pk(n) holds under a finite positive linear-contact hypothesis.
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BibTeX
@misc{li2026modelthreshold,
title = {{A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $σ_k$-curvature Metrics}},
author = {Jiahuan Li and Yilu Liu and Xi-Nan Ma},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.16571},
eprint = {2608.16571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16571},
}
RIS
TY - UNPB
TI - A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive σ_k-curvature Metrics
AU - Jiahuan Li
AU - Yilu Liu
AU - Xi-Nan Ma
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.16571
UR - https://arxiv.org/abs/2608.16571
AB - Let k∈ℕ satisfy 1<k<n/2, and let Σ^p⊂Sn^n be a closed smooth embedded submanifold. We prove that a complete conformal metric g=v^−2g_Sn^n on Sn^n∖Σ satisfying λ(g^−1A_g)∈Gk and σ_k(g^−1A_g)=κ>0 must obey p≤ p_k(n), where p_k is the model threshold determined by Hh^p+1×Sn^n−p−1. When k=2 and n=m^2, we construct a smooth complete equality example on Sn^n∖Sn^(m^2−m−2)/2. We also prove that the strict inequality p<p_k(n) holds under a finite positive linear-contact hypothesis.
ER -