Extreme local extrema of the sine-Gordon field

We prove that for $\beta<6\pi$ the local extremal process of the massive sine-Gordon field on the unit torus in $d=2$ converges to a Poisson point process with random intensity measure ${\rm Z}^{\mathrm{SG}}(dx) \otimes e^{-\alpha h}dh$ for some $\alpha>0$. The proof combines existing methods for the extremal process associated to the Gaussian free field, which was introduced and studied by Biskup and Louidor, and a strong coupling between the sine-Gordon field and the Gaussian free field.