Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle

The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure.  In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make