In ordinary fluid dynamics — the Euler equations without surface tension — thick vortex rings exist as steady solutions. A toroidal region of concentrated vorticity carries its own velocity field, propagating through the fluid indefinitely. The mathematics has been settled for decades: these structures are solutions of a nonlinear elliptic equation, and they exist across a range of ring geometries from thin filaments to fat, nearly spherical tori.
Now add surface tension. Replace the vortex ring with a bubble ring — a toroidal gas cavity surrounded by liquid. The question: can a thick bubble ring persist as a steady solution of the capillary Euler equations at low Weber numbers, where surface tension dominates?
The paper (arXiv:2603.24217, March 2026) proves the answer is no. While spheroidal bubbles approach a sphere in the low Weber number limit — surface tension smooths everything — the thick ring topology is not smoothable. The free-boundary capillary equations simply do not support thick bubble rings when surface tension is strong. The structure exists in the ordinary equations but is destroyed by the boundary condition that the capillary equations impose.
The mechanism is clean: surface tension penalizes curvature. A thick torus has regions of both positive and negative Gaussian curvature — convex on the outside, saddle-shaped on the inside. The pressure balance required by the Laplace equation across a free boundary cannot be satisfied simultaneously in both regions while maintaining a stationary toroidal shape. The topology that curvature-free fluid dynamics permits becomes topologically impossible when curvature is penalized.
Vortex rings survive because they are internal structures — no free boundary, no surface tension, no curvature penalty. Bubble rings require a boundary. The boundary is what kills them.