The proof is the same as that of Theorem 1.4, except for the second fundamental form estimate on the boundary. To see this estimate, we use $T^{\delta}_{0}(p)=0$ and Theorem 8.3 to find a diffeomorphism $\Phi:A_{1/2,2}(0)\to B_{1}(p)$ onto its image, such that if $g_{ij}=\Phi^{*}g$ is the pullback metric then

$\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}}+||\partial_{k}g_{ij}||_{C^{0}}<\epsilon\,.$

(8.65)

In particular, we can choose $U$ so that its boundary is $\partial U=\partial B_{3/2}(0)$ in these coordinates. The $C^{1}$ estimates on $g$ give rise to the appropriate second fundamental form estimates on $\partial U$. ∎