  According to the positive energy conjecture of Horowitz and Myers, there is a
specific supergravity solution, AdS soliton, which has minimum energy among all
asymptotically locally AdS solutions with the same boundary conditions. Related
to the issue of semiclassical stability of AdS soliton in the context of pure
gravity with a negative cosmological constant, physical boundary conditions are
determined for an instanton solution which would be responsible for vacuum
decay by barrier penetration. Certain geometric properties of instantons are
studied, using Hermitian differential operators. On a $d$-dimensional
instanton, it is shown that there are $d-2$ harmonic functions. A class of
instanton solutions, obeying more restrictive boundary conditions, is proved to
have $d-1$ Killing vectors which also commute. All but one of the Killing
vectors are duals of harmonic one-forms, which are gradients of harmonic
functions, and do not have any fixed points.
