  We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0,
u_1) whose energy is smaller than that of (W,0), where W is the well-known
radial positive solution to the corresponding ellipyic equation, the following
dichotomy holds: a) if the homogeneous Sobolev norm H^1 of u_0 is smaller than
that of W, we have global well-posedness and scattering, b) if the homogeneous
Sobolev norm H^1 of u_0 is larger than that of W, there is blow-up in finite
time. Our general approach is the one we introduced in our previous work on the
corresponding problem for NLS (math.AP/0610266, Inventiones Math 2006, Online
First), where we proved the corresponding result for NLS in the radial case. In
the case of the wave equation we are able to treat general data by using a
further conservation law in the energy space, the finite speed of propagation
and Lorentz transformations to establish a crucial orthogonality property for "
energy critical " elements. To prove the required rigidity theorem in the case
of blow-up in finite time, we cannot use the invariance of the L^2 norm as in
the case of NLS. Instead, (following earlier work of Merle-Zaag and of
Giga-Kohn in the parabolic case) we introduce self-similar variables. We thus
find a further Liapunov function which allows us to reduce matters to a
degenerate elliptic problem with critical non-linearity. We use unique
continuation to rule out the existence of non-zero solutions for the degenerate
elliptic problem, thus completing the proof.
