  We provide a complete local well-posedness theory in $H^s$ based Sobolev
spaces for the free boundary incompressible Euler equations with zero surface
tension on a connected fluid domain. Our well-posedness theory includes: (i)
Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness,
and the first proof of continuous dependence on the data, all in low regularity
Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the
level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$
regularity of the free surface; (iii) Stability bounds: We construct a
nonlinear functional which measures, in a suitable sense, the distance between
two solutions (even when defined on different domains) and we show that this
distance is propagated by the flow; (iv) Energy estimates: We prove refined,
essentially scale invariant energy estimates for solutions, relying on a newly
constructed family of elliptic estimates; (v) Continuation criterion: We give
the first proof of a sharp continuation criterion in the physically relevant
pointwise norms, at the level of scaling. In essence, we show that solutions
can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free
surface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as the
Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the
construction of regular solutions.
  Our entire approach is in the Eulerian framework and can be adapted to work
in more general fluid domains.
