In terms of the original coframes $\{dx,dy,dz,\theta\}$, the volume form is given by

(7.122)

$$\dvol_{g_{j}}=V_{\beta_{j}}dx\wedge dy\wedge dz\wedge\theta.$$

By definition,

(7.123)

$\displaystyle*(dx)=dy\wedge dz\wedge\theta,\ *(dy)=-dx\wedge dz\wedge\theta,\ *(dz)=dx\wedge dy\wedge\theta,$

which implies

(7.124)

$$\Delta_{g_{j}}x=\Delta_{g_{j}}y=\Delta_{g_{j}}z=0.$$

After rescaling, we have that

(7.125)

$$\Delta_{\tilde{g}_{j}}x_{j}=\Delta_{\tilde{g}_{j}}y_{j}=\Delta_{\tilde{g}_{j}}z_{j}=0.$$

By Lemma 7.11, the pointwise gradient estimate holds,

(7.126)

$$|\nabla x_{j}|_{\tilde{g}_{j}}=|\nabla y_{j}|_{\tilde{g}_{j}}=|\nabla z_{j}|_{\tilde{g}_{j}}\to 1.$$

Now we estimate the Hessian of the harmonic functions $x_{j}$, $y_{j}$ and $z_{j}$. It suffices to check it for $x_{j}$. First, Bochner’s formula gives that

(7.127)

$$\frac{1}{2}\Delta_{\tilde{g}_{j}}|\nabla x_{j}|_{\tilde{g}_{j}}^{2}=|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}.$$

Due to Cheeger-Colding (see [CC96]), there exist cutoff functions $\varphi_{j}:\mathcal{M}\to[0,1]$ with

(7.128)

$\displaystyle\varphi_{j}(x)=\begin{cases}1,\ x\in B_{R}(p_{j}),\\ 0,\ x\in\mathcal{M}\setminus B_{2R}(p_{j})\end{cases}$

and there exists an absolute constant $C_{0}>0$ such that

(7.129)

$$R|\nabla_{\tilde{g}_{j}}\varphi_{j}|_{\tilde{g}_{j}}+R^{2}|\Delta_{\tilde{g}_{j}}\varphi_{j}|\leq C_{0}.$$

Integrating (7.127) over $B_{4R}(p_{j})$,

(7.130)

$\displaystyle\begin{split}\fint_{B_{4R}(p_{j})}\varphi_{j}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\dvol_{\tilde{g}_{j}}&=\frac{1}{\Vol_{\tilde{g}_{j}}(B_{4R}(p_{j}))}\int_{B_{4R}(p_{j})}\varphi_{j}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\dvol_{\tilde{g}_{j}}\\ &=\frac{1}{\Vol_{\tilde{g}_{j}}(B_{4R}(p_{j}))}\int_{B_{4R}(p_{j})}\frac{1}{2}\varphi_{j}\Delta_{\tilde{g}_{j}}(|\nabla x_{j}|_{\tilde{g}_{j}}^{2}-1)\dvol_{\tilde{g}_{j}}\\ &=\frac{1}{2\Vol_{\tilde{g}_{j}}(B_{4R}(p_{j}))}\int_{B_{4R}(p_{j})}(\Delta_{\tilde{g}_{j}}\varphi_{j})\cdot(|\nabla x_{j}|_{\tilde{g}_{j}}^{2}-1)\dvol_{\tilde{g}_{j}}\rightarrow 0,\end{split}$

as $j\rightarrow\infty$. Therefore, by volume comparison,

(7.131)

$$\fint_{B_{R}(p_{j})}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\dvol_{\tilde{g}_{j}}\to 0,$$

as $j\rightarrow\infty$. Let $p_{\infty}\in\mathbb{R}^{3}\setminus\{0^{3}\}$ with $B_{2\bar{s}_{0}}(p_{\infty})\subset\mathbb{R}^{3}\setminus\{0^{3}\}$ and we choose a sequence of geodesic balls $B_{2\bar{s}_{0}}(p_{j})$ such that

(7.132)

$$(B_{2\bar{s}_{0}}(p_{j}),\tilde{g}_{j})\xrightarrow{GH}(B_{2\bar{s}_{0}}(p_{\infty}),g_{0}).$$

By Lemma 7.7, the curvatures on $B_{\bar{s}_{0}}(p_{j})$ are uniformly bounded by $C\cdot\bar{s}_{0}^{-2}$ and $C>0$ is an absolute constant. On the other hand, since $\Delta_{\tilde{g}_{j}}x_{j}=0$, (7.131) can be strengthened to

(7.133)

$$\sup\limits_{B_{\bar{s}_{0}}(p_{j})}|\nabla^{2}x_{j}|_{\tilde{g}_{j}}^{2}\to 0.$$

The proof is done.

∎