Global Bifurcation and the Constructive Existence of Overhanging Periodic Steady Water Waves

We provide a constructive existence proof for overhanging periodic gravity water waves with constant vorticity. By introducing a conformal mapping formulation, we parameterize the fluid domain to accommodate multi-valued surface heights, bypassing coordinate singularities that arise in traditional domain-flattening frameworks. We operate at fixed, macroscopic physical parameters and employ a global bifurcation framework, supplemented by a computer-assisted proof based on the Newton-Kantorovich theorem, to constructively prove the existence of the branch of exact solutions. Crucially, this allows us to provide the first rigorous existence proof of overhanging periodic gravity water waves obtained along a finite-depth global bifurcation branch from the flat state at fixed $O(1)$ physical parameters. Moreover, our analysis confirms the existence of a global solution branch that transitions to overhanging profiles and resolves the conjecture of Constantin, Strauss, and Varvaruca regarding the topological termination of this branch via physical self-intersection at the trough line.