Global bifurcation of doubly periodic gravity-capillary waves on Beltrami flows

We prove the existence of a global family of steady, doubly periodic gravity-capillary waves on Beltrami flows. This is the first rigorous existence result for genuinely three-dimensional surface waves, with or without vorticity, beyond the perturbative regime close to simple explicit solutions. The proof is based on reformulating the steady water wave problem as a bifurcation problem of the form `identity plus compact' and applying a global bifurcation argument in H\"older spaces. The main challenge is that the kernel of the linearisation at the bifurcation point is two-dimensional, and that both kernel elements are necessary to obtain genuinely three-dimensional solutions. Since this prevents the use of classical global bifurcation theory, we introduce a novel reformulation of the bifurcation problem using the parameterisation of a local family of solutions bifurcating from laminar flow. In this new parameter space, we apply a variation of analytic global bifurcation theory. Along the branch, we then prove a sharper blow-up alternative, namely blow-up of the surface gradient in $C^{0,\gamma}$.