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We study the following problem: Given initial data on a compact Cauchyhorizon, does there exist a unique solution to wave equations on the globallyhyperbolic region? Our main results apply to any spacetime satisfying the nullenergy condition and containing a compact Cauchy horizon with surface gravitythat can be normalised to a non-zero constant. Examples include the Misnerspacetime and the Taub-NUT spacetime. We prove an energy estimate close to theCauchy horizon for wave equations acting on sections of vector bundles. Usingthis estimate we prove that if a linear wave equation can be solved up to anyorder at the Cauchy horizon, then there exists a unique solution on theglobally hyperbolic region. As a consequence, we prove several existence anduniqueness results for linear and non-linear wave equations without assuminganalyticity or symmetry of the spacetime and without assuming that thegenerators close. We overcome in particular the essential remaining difficultyin proving that vacuum spacetimes with a compact Cauchy horizon with constantnon-zero surface gravity necessarily admits a Killing vector field. This workis therefore related to the strong cosmic censorship conjecture.

Wave equations with initial data on compact Cauchy horizons