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

Abstract

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