Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves

This paper is concerned with a priori C ∞ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a Diophantine condition are automatically C ∞ . In particular, we prove that the solutions defined by Iooss and Plotnikov are C ∞ . Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.