  The exploration of shape metamorphism, surface reconstruction, and image
interpolation raises fundamental inquiries concerning the $C^1$ and
higher-order regularity of $\infty$-harmonic potentials -- a specialized
category of $\infty$-harmonic functions. Additionally, it prompts questions
regarding their corresponding approximations using $p$-harmonic potentials. It
is worth noting that establishing $C^1$ and higher-order regularity for
$\infty$-harmonic functions remains a central concern within the realm of
$\infty$-Laplace equations and $L^\infty$-variational problems.
  In this study, we investigate the regularity properties from $p$-harmonic
potentials to $\infty$-harmonic potentials within arbitrary convex ring domains
$\Omega=\Omega_0\backslash \overline \Omega_1$ in $\mathbb R^n$. Here
$\Omega_0$ is a bounded convex domain in $\mathbb R^n$ and
$\overline\Omega_1\subset \Omega_0$ is a compact convex set. We prove the
interior $C^1$ and some Sobolev regularity for $\infty$-harmonic potentials.
