Compactness Theorem of Complete k-Curvature Manifolds with Isolated Singularities

Abstract

In this paper we prove that the set of metrics conformal to the standard metric on ${\mathbb{S}}^n\backslash \{p_1,\cdots ,p_l\}$ is locally compact in $C^{m,\alpha }$ topology for any $m>0$, whenever the metrics have constant ${\sigma }_k$ curvature and the $k$-Dilational Pohozaev invariants have positive lower bound for $k<n/2$. Here the $k$-Dilational Pohozaev invariants come from the Kazdan-Warner type identity for the ${\sigma }_k$ curvature, which is derived by Viaclovsky \cite{Viac2000} and Han \cite{H1}. When $k=1$, Pollack \cite{Pollack} proved the compactness results for the complete metrics of constant positive scalar curvature on ${\mathbb{S}}^n\backslash \{p_1,\cdots ,p_l\}$.

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