  We establish a new uniqueness theorem for the three dimensional
Schwarzschild-de Sitter metrics. For this some new or improved tools are
developed. These include a reverse Lojasiewicz inequality, which holds in a
neighborhood of the extremal points of any smooth function. We further prove
smoothness of the set of maxima of the lapse, whenever this set contains a
topological hypersurface. This leads to a new strategy for the classification
of well behaved static solutions of Einstein equations with a positive
cosmological constant, based on the geometry of the maximum-set of the lapse.
