  We prove the existence and uniqueness of solutions to the time-dependent
incompressible Navier-Stokes equations with a free-boundary governed by surface
tension. The solution is found using a topological fixed-point theorem for a
nonlinear iteration scheme, requiring at each step, the solution of a model
linear problem consisting of the time-dependent Stokes equation with linearized
mean-curvature forcing on the boundary. We use energy methods to establish new
types of spacetime inequalities that allow us to find a unique weak solution to
this problem. We then prove regularity of the weak solution, and establish the
a priori estimates required by the nonlinear iteration process.
