Classification of finite-time blow-up of strong solutions to the incompressible free boundary Euler equations with surface tension

We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary. Uniquely, we make $\mathit{\text{no}}$ assumptions on symmetry, periodicity, graph representation, or domain topology (simple connectivity). At the maximal existence time $T<\mathrm{\infty }$, up to which the velocity field and the free boundary can be continued in $H^3\times H^4$, blow-up must occur in at least one of five mutually exclusive ways: (i) self-intersection of the free boundary for the first time; (ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or the free boundary regularity in $H^{2+\epsilon }$ (for any sufficiently small constant $\epsilon >0$); (iii) loss of $H^{\frac{5}{2}}$ regularity for the normal boundary velocity; (iv) the $L_t^1{L}^{\mathrm{\infty }}$-blow-up of the tangential velocity gradient on the boundary; or (v) the $L_t^1{L}^{\mathrm{\infty }}$-blow-up of the full velocity gradient in the interior. Furthermore, for simply connected domains, blow-up scenario (v) simplifies to a vorticity-based Beale-Kato-Majda criterion, and in particular, irrotational flows admit blow-up only at the free boundary.