Large floating viscous bubbles whose interior gas is rapidly depressurized exhibit a fascinating instability, whereby radial wrinkles permeate the liquid film in the course of its flattening (Debregeas et al, Science 1998; DaSilviera et al., Science 2000, Oratis et al., Science 2020). We show that this instability emerges from a largely unexplored type of classical hydrodynamics, that is geometrically-nonlinear Stokes flow of curved, volumetrically-incompressible films. This theoretical framework highlights profound similarities and differences between the mechanics of elastic sheets and viscous films, revealing the experimental observations of Oratis et al. as a universal, curvature-driven surface dynamics, imparted by viscous resistance to temporal variations of the surface's Gaussian curvature. This novel surface dynamics has close ties to the kinetics of first-order phase transitions and to ``Jelium physics" in continuum media, where topological defects, akin to charges in electrostatic media, spontaneously emerge to screen elastic stresses.