A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive ${\sigma }_k$-curvature Metrics

Let $k\in \mathbb{N}$ satisfy $1<k<n\mathrm{/}2$, and let ${\mathrm{\Sigma }}^p\subset {\text{Sn}}^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}{g}_{{\text{Sn}}^n}$ on ${\text{Sn}}^n\setminus \mathrm{\Sigma }$ satisfying $\lambda (g^{-1}{A}_g)\in \text{Gk}$ and ${\sigma }_k(g^{-1}{A}_g)=\kappa >0$ must obey $$p\le p_k(n),$$ where $p_k$ is the model threshold determined by ${\text{Hh}}^{p+1}\times {\text{Sn}}^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on ${\text{Sn}}^n\setminus {\text{Sn}}^{(m^2-m-2)\mathrm{/}2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.