  We establish a black hole uniqueness theorem for Schwarzschild-de Sitter
spacetime, also called Kottler spacetime, which satisfies Einstein's field
equations of general relativity with positive cosmological constant. Our result
concerns the class of static vacuum spacetimes with compact spacelike slices
and regular maximal level set of the lapse function. We provide a
characterization of the interior domain of communication of the Kottler
spacetime, which surrounds an inner horizon and is surrounded by a cosmological
horizon. The proposed proof combines arguments from the theory of partial
differential equations and differential geometry, and is centered on a detailed
study of a possibly singular foliation. We also apply our technique in the
Riemannian setting, and establish the validity of the so-called Besse
conjecture.
