Properness for scaled gauged maps

We prove properness of moduli stacks of gauged maps satisfying a stability condition introduced by Mundet [40], Schmitt [46] and Ziltener [57]. The proof combines a git construction of Schmitt [46], properness for twisted stable maps by Abramovich-Vistoli [1], a variation of a boundedness argument due to Ciocan-Fontanine-Kim-Maulik [13], and a removal of singularities for bundles on surfaces in Colliot-Thélène-Sansuc [14].