Let the cylinder $S^{n-1}\times \mathbb R$ be equipped with the standard metric $g$. Suppose that there exists a sequence of metrics $g_{\epsilon}$ on $S^{n-1}\times \mathbb R$ such that $g_{\epsilon}$ converge smoothly to $g$ in the Cheeger-Gromov sense, can we prove that if $\epsilon$ is sufficiently small, any point $p$ in $S^{n-1}\times \mathbb [-1,1]$ lies on a unique sphere $S^{n-1}$ with constant mean curvature w.r.t. $g_{\epsilon}$?


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