We construct gravity water waves with constant vorticity having the approximate shape of a disk joined to a strip by a thin neck. This is the first rigorous existence result for such waves, which have been seen in numerics since the 80s and 90s. Our method is related to the construction of constant mean curvature surfaces through gluing, and involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an “exceptional domain” discovered by Hauswirth, Hélein, and Pacard.

This is joint work with Juan Dávila, Manuel del Pino, and Monica Musso.