Recall that $(M_{k}, g_k,O_k)$ of complete pointed Riemannian manifolds converges smoothly in the sense of Cheeger-Gromov to $(M_{\infty}, g_{\infty}, O_{\infty})$ if there exists an exhaustion of open sets $U_k$ of $M_{\infty}$ containing $ O_{\infty}$, and a sequence of diffeomorphism $\phi_k$ from $U_k$ to $V_k=\phi_k(U_k) \subset M_k$ with $\phi_k (O_{\infty})=O_k$, such that the pull back $(U_k, \phi_k^{\ast} (g_k))$ converges to $(M_{\infty}, g_{\infty})$ uniformly on compact sets in $M_{\infty}$.
These days I am interested in how wild the diffeomorphism $\phi_k$ can be. Consider a sequence of rotationaly invariant metrics on $\mathbb{R}^n$ with uniformly bounded geometry and injective radius at the origin has a uniform lower bound (which can be ensured by for example, choosing a uniform bounded positive cone angle at infinity for the sequence metrics), fix the sequence of points to be the origin, by the compactness theorem we expect a limit metric which is still rotationaly invariant. However, I am not sure if the limit is necessarily rotationaly invariant. If so does it mean that in this special case, we can choose the sequence of diffeomorphism to be a sequence of rotations around the origin?
I understand that a way to think about this symmetric case is to go over the general construction on diffeomorphism and the limit metric in the proof of Compactness Theorem. It is a little involved and I am still on the way to understand it. Any suggestions or help will be appreciated.
Added after posted:
To make the point of "understand the diffeomorphism $\phi_k$" clear, let me continue on the rotationally symmetric example, Let us put one more restrition on the sequence $(M_k, g_k, O_k)=(R^n, g_k,O)$ that the scalar curvature $R(O)$ attain the maximum for each $g_k$, then after the taking limit we can get a metric $(M_\infty, g_\infty,O_\infty)$ which also have $R(O_\infty)$ as a maximum on $M_\infty$, But if we take $O_k$ to be a sequence points $P_k$ suitably close to the origin $O$, the limit will be a new metric $(N_\infty, h_\infty,P_\infty)$. It is reasonably to believe the new metric could be the same as $(M_\infty, g_\infty,O_\infty)$. However $R(P_\infty)$ might not be the maximum of scalar curvature on $N_\infty$ in general, So it is natural to find a maximum of scalar curvature on $N_\infty$ first, name it $Q_\infty \in N_\infty$, then check how far the pull back sequence $\Phi_k(Q_\infty) \in M_k$ is from $O$. It sounds to me that it is important to understand the behavior of $\Phi_k$. However I feel that it is a little wild even for this example.
