Proceeding from a recent construction (Fraenkel 2007, Arch. Rational Mech. Anal. 183, 187–214), this paper contains three contributions to the theory of the extreme gravity wave on water (or “wave of greatest height”) initiated by Stokes in 1880. The first is a solution, of the extreme form of the integral equation due to Nekrasov, that consists of an explicit function plus a (rigorously) estimated remainder; the remainder vanishes at the crest and at the trough of the wave and in between has a maximum value less than 0.0026 times the maximum value of the explicit part. The second contribution establishes local uniqueness for a variant of the Nekrasov equation; we prove that this equation has only one solution between upper and lower bounding functions that are moderately far apart. The third contribution consists of a table and graph of strict upper and lower bounds for the interface between air and water in the physical plane.
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