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In dimension 2, we introduce a distributional Jacobian determinant $\detDV_\beta(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto|Dv|^\beta(v_{x_1},-v_{x_2})$ for any $\beta>-1$, whenever $v\in W^{1,2}_{\text{loc}}$ and $\beta |Dv|^{1+\beta}\in W^{1,2}_{\text{loc}}$. Then forany planar $\infty$-harmonic function $u$, we show that such distributionalJacobian determinant is a nonnegative Radon measure with some quantitativelocal lower and upper bounds. We also give the following two applications.  (i) Applying this result with $\beta=0$, we develop an approach to build up aLiouville theorem, which improves that of Savin [33]. Precisely, if $u$ is$\infty$-harmonic functions in whole ${\mathbb R}^2$ with $$\liminf_{R\to\infty}\inf_{c\in\mathbb R}\frac1{R^3}\int_{B(0,R)}|u(x)-c|\,dx<\infty,$$ then $u=b+a\cdot x$ for some$b\in{\mathbb R}$ and $a\in{\mathbb R}^2$.  (ii) Denoting by $u_p$ the $p$-harmonic function having the same nonconstantboundary condition as $u$, we show that $\det DV_\beta(Du_p) \to \detDV_\beta(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radonmeasure. Recall that $V_\beta(Du_p)$ is always quasiregular mappings, but$V_\beta(Du)$ is not in general.

Jacobian determinants for (nonlinear) gradient of planar $\infty$-harmonic functions and applications