Convexity of Stokes Waves of Extreme Form

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The trajectories of particles in Stokes waves (2006, background): “The substantial amount of analytical theory made available mainly through the work of Toland and collaborators (see [1, 2, 4, 27 , 34, 35] and citations therein) unveils to a large extent the fascinating structure of this classical hydrodynamical problem.”
A geometric approach to generalized Stokes conjectures (2009, background): “Moreover, Plotnikov and Toland proved convexity of the two branches of the free surface [21].”
Pressure beneath a Stokes wave (2009, background): “However, it is known [2, 24] that (2)x < 1 so that the last two relations imply (3.”