ScalingStacks

Theorem 5.6. (Smooth convergence in generic regions) As $s\to+\infty$ along the subsequence, assume the coordinate ball $B(x,2r(x))\subset\mathcal{R}$, then on the region $(s^{-1}\text{Log})^{-1}(B(x,r(x)))\subset X_{s}$, we have the following higher regularity estimates with respect to the $C^{k,\gamma}$-norm in the $s^{-1}\log z^{m_{i}}$ coordinates.

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  In the face type region $U_{w}^{s,face}$ case

  $$\left\lVert\varphi_{CY,s,0}-u_{\infty}\circ s^{-1}\text{Log}\right\rVert_{C^{k,\gamma}((s^{-1}\text{Log})^{-1}(B(x,r(x)))}\to 0.$$

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  In the star type region $U_{w}^{s,*}$ case, for $\langle m,w\rangle=1$,

  $$\left\lVert\varphi_{CY,s,m}-u_{\infty,m}\circ s^{-1}\text{Log}\right\rVert_{C^{k,\gamma}((s^{-1}\text{Log})^{-1}(B(x,r(x)))}\to 0.$$

The convergence rate is uniform for $x$ on any fixed compact subset of $\mathcal{R}$.