Rigidity of symmetric doubly-periodic water waves near shear flows

We prove that symmetric, doubly periodic, capillary-gravity water waves in finite depth bifurcating from non-uniform non-stagnant shear flows are necessarily two-dimensional to leading order. This is in stark contrast to the case of uniform background flows, where the existence of truly three-dimensional bifurcating waves is well known. While this paper focuses on capillary-gravity water waves, our proof also applies to other suitable types of dynamic boundary conditions, such as hydroelastic waves.