Convexity of Stokes Waves of Extreme Form

Existence is established of a piecewise-convex, periodic, planar curve S below which is defined a harmonic function which simultaneously satisfies prescribed Dirichlet and Neumann boundary conditions on S. In hydrodynamics this corresponds to the existence of a periodic Stokes wave of extreme form which has a convex profile between consecutive stagnation points where there is a corner with a contained angle of 120°.