  For a two-surface B tending to an infinite--radius round sphere at spatial
infinity, we consider the Brown--York boundary integral H_B belonging to the
energy sector of the gravitational Hamiltonian. Assuming that the lapse
function behaves as N \sim 1 in the limit, we find agreement between H_B and
the total Arnowitt--Deser--Misner energy, an agreement first noted by Braden,
Brown, Whiting, and York. However, we argue that the Arnowitt--Deser--Misner
mass--aspect differs from a gauge invariant mass--aspect by a pure divergence
on the unit sphere. We also examine the boundary integral H_B corresponding to
the Hamiltonian generator of an asymptotic boost, in which case the lapse N
\sim x^k grows like one of the asymptotically Cartesian coordinate functions.
Such an integral defines the kth component of the center of mass for a Cauchy
surface \Sigma bounded by B. In the large--radius limit, we find agreement
between H_B and an integral introduced by Beig and O'Murchadha. Although both
H_B and the Beig--O'Murchadha integral are naively divergent, they are in fact
finite modulo the Hamiltonian constraint. Furthermore, we examine the
relationship between H_B and a certain two--surface integral linear in the
spacetime Riemann curvature tensor. Similar integrals featuring the curvature
appear in works by Ashtekar and Hansen, Penrose, Goldberg, and Hayward. Within
the canonical 3+1 formalism, we define gravitational energy and
center--of--mass as certain moments of Riemann curvature.
