  We prove the linear stability of slowly rotating Kerr black holes as
solutions of the Einstein vacuum equation: linearized perturbations of a Kerr
metric decay at an inverse polynomial rate to a linearized Kerr metric plus a
pure gauge term. We work in a natural wave map/DeTurck gauge and show that the
pure gauge term can be taken to lie in a fixed 7-dimensional space with a
simple geometric interpretation. Our proof rests on a robust general framework,
based on recent advances in microlocal analysis and non-elliptic Fredholm
theory, for the analysis of resolvents of operators on asymptotically flat
spaces. With the mode stability of the Schwarzschild metric as well as of
certain scalar and 1-form wave operators on the Schwarzschild spacetime as an
input, we establish the linear stability of slowly rotating Kerr black holes
using perturbative arguments; in particular, our proof does not make any use of
special algebraic properties of the Kerr metric. The heart of the paper is a
detailed description of the resolvent of the linearization of a suitable
hyperbolic gauge-fixed Einstein operator at low energies. As in previous work
by the second and third authors on the nonlinear stability of cosmological
black holes, constraint damping plays an important role. Here, it eliminates
certain pathological generalized zero energy states; it also ensures that
solutions of our hyperbolic formulation of the linearized Einstein equation
have the stated asymptotics and decay for general initial data and forcing
terms, which is a useful feature in nonlinear and numerical applications.
