  We prove two theorems, announced in hep-th/0108170, for static spacetimes
that solve Einstein's equation with negative cosmological constant. The first
is a general structure theorem for spacetimes obeying a certain convexity
condition near infinity, analogous to the structure theorems of Cheeger and
Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with
Ricci-flat conformal boundary, the convexity condition is associated with
negative mass. The second theorem is a uniqueness theorem for the negative mass
AdS soliton spacetime. This result lends support to the new positive mass
conjecture due to Horowitz and Myers which states that the unique lowest mass
solution which asymptotes to the AdS soliton is the soliton itself. This
conjecture was motivated by a nonsupersymmetric version of the AdS/CFT
correspondence. Our results add to the growing body of rigorous mathematical
results inspired by the AdS/CFT correspondence conjecture. Our techniques
exploit a special geometric feature which the universal cover of the soliton
spacetime shares with familiar ``ground state'' spacetimes such as Minkowski
spacetime, namely, the presence of a null line, or complete achronal null
geodesic, and the totally geodesic null hypersurface that it determines. En
route, we provide an analysis of the boundary data at conformal infinity for
the Lorentzian signature static Einstein equations, in the spirit of the
Fefferman-Graham analysis for the Riemannian signature case. This leads us to
generalize to arbitrary dimension a mass definition for static asymptotically
AdS spacetimes given by Chru\'sciel and Simon. We prove equivalence of this
mass definition with those of Ashtekar-Magnon and Hawking-Horowitz.
