常正σ_k曲率度量的光滑奇异集的模型阈值维数界与尖锐临界端点
A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $σ_k$-curvature Metrics
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中文总结 AI 辅助
针对常正σ_k曲率度量的光滑奇异集，该研究推导了其维数的模型阈值界，构造了特定参数下的等例，并在有限正线性接触假设下证明了严格不等式。
中文摘要 AI 辅助
设k为自然数且满足1<k<n/2，Σ^p⊂S^n是闭光滑嵌入子流形。我们证明：S^n\\Σ上满足λ(g⁻¹A_g)∈Γ_k且σ_k(g⁻¹A_g)=κ>0的共形度量g=v⁻²g_{S^n}必满足p≤p_k(n)，其中p_k是由H^{p+1}×S^{n-p-1}确定的模型阈值。当k=2且n=m²时，我们在S^n\S^{(m²-m-2)/2}上构造了光滑完全等例，还证明在有限正线性接触假设下严格不等式p<p_k(n)成立。
英文摘要
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 $σ_k(g^{-1}A_g)=κ>0$ 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.
