# Foliation of cylinders by constant mean curvature spheres

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. $$g_{\epsilon}$$?