  We study the following problem: Given initial data on a compact Cauchy
horizon, does there exist a unique solution to wave equations on the globally
hyperbolic region? Our main results apply to any spacetime satisfying the null
energy condition and containing a compact Cauchy horizon with surface gravity
that can be normalised to a non-zero constant. Examples include the Misner
spacetime and the Taub-NUT spacetime. We prove an energy estimate close to the
Cauchy horizon for wave equations acting on sections of vector bundles. Using
this estimate we prove that if a linear wave equation can be solved up to any
order at the Cauchy horizon, then there exists a unique solution on the
globally hyperbolic region. As a consequence, we prove several existence and
uniqueness results for linear and non-linear wave equations without assuming
analyticity or symmetry of the spacetime and without assuming that the
generators close. We overcome in particular the essential remaining difficulty
in proving that vacuum spacetimes with a compact Cauchy horizon with constant
non-zero surface gravity necessarily admits a Killing vector field. This work
is therefore related to the strong cosmic censorship conjecture.
