Working on some $Q$-curvature equation in dimension $4$, I have been faced with singular metric of the form $(\mathbb{B}, e^{-1/\vert x\vert ^2} \vert dx\vert)$. I try to figure out to what those metrics look like. In order to do so, I refer to the well known $2$-dimensional where we have conical metric $(\mathbb{D}, \vert x\vert^{\beta-1} \vert dx\vert)$ which are isometric to a cone of angle $2\pi \beta$. If $\beta=n$ is an integer, I can think about it as the neighborhood of $0$ of and $n$-fold branched unit disc. My question is: is there is a similar interpretation for $(\mathbb{D}, e^{-1/\vert x\vert ^2} \vert dx\vert)$ (in dimension 2)? Can we think about this metric as the limit of some more classical degenerate metric? Any reference will be welcome. Thx in advance