Minimal surfaces in $q$-deformed AdS$_5\times$S$^5$ with Poincare coordinates

We study minimal surfaces in $q$-deformed AdS$_5\times$S$^5$ with a new coordinate system introduced in the previous work 1408.2189. In this letter, we introduce Poincare coordinates for the deformed theory. Then we construct minimal surfaces whose boundary shape is a circle. The solution corresponds to a 1/2 BPS circular Wilson loop in the $q\to 1$ limit. A remarkable point is that the classical Euclidean action is not divergent unlike the original one. This finiteness indicates that the $q$-deformation may be regarded as a UV regularization.