  An approximate Killing field may be defined on a compact, Riemannian geometry
by solving an eigenvalue problem for a certain elliptic operator. This paper
studies the effect of small perturbations in the Riemannian metric on the
resulting vector field. It shows that small metric perturbations, as measured
using a Sobolev-type supremum norm on the space of Riemannian geometries on a
fixed manifold, yield small perturbations in the approximate Killing field, as
measured using a Hilbert-type square integral norm. It also discusses
applications to the problem of computing the spin of a generic black hole in
general relativity.
