For every $\epsilon>0$, there exists $\delta=\delta(n,\epsilon)$, such that the following holds. If $M^{n}_{1},M^{n}_{2}$ are Riemannian manifolds and $U_{j}\subset M_{j}$ are subsets such that $r_{h}(x)>r>0$ for each $x\in U_{j}$, and

$$d_{GH}(B_{r}(U_{1}),B_{r}(U_{2}))<\epsilon r\,,$$

then there exist open sets $B_{r/2}(U_{j})\subseteq U^{\prime}_{j}\subseteq B_{r}(U_{j})$ and a $C^{2}$ diffeomorphism $\Phi:U^{\prime}_{1}\to U^{\prime}_{2}$, such that

$\displaystyle||g_{1}-\Phi^{*}g_{2}||_{C^{0}}<\epsilon\,.$

(8.1)

If we further assume $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$, $j=1,2$, then $\Phi$ is in $C^{2,\alpha}\cap W^{3,q}$ for all $\alpha<1$ and $q<\infty$, and in harmonic coordinates on $U^{\prime}_{1}$ we have

$\displaystyle||g_{1}-\Phi^{*}g_{2}||_{C^{0}}+r^{1+\alpha}||\partial_{i}\Phi^{*}g_{2}||_{C^{\alpha}}+r^{2}||\partial_{i}\partial_{j}\Phi^{*}g_{2}||_{L^{q}}\leq C(n,\alpha,q)\epsilon\,.$

(8.2)