The zero surface tension limit of three-dimensional water waves

We establish that the limit of the water wave with surface tension, as surface tension vanishes, is the water wave without surface tension. The main tool is an energy estimate which is uniform in the surface tension parameter. Before establishing estimates, we reformulate the problem using suitable variables and an isothermal parameterization. With these variables and parameterizations, estimates for the water wave with or without surface tension are straightforward. In particular, this provides a new proof of existence of irrotational water waves in three space dimensions.