An isoperimetric comparison theorem for Schwarzschild space and other manifolds

Abstract

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric ( n − 1 ) (n-1) -spheres of a spherically symmetric n n -manifold are isoperimetric hypersurfaces, meaning that they minimize ( n − 1 ) (n-1) -dimensional area among hypersurfaces enclosing the same n n -volume. This result greatly generalizes the result of Bray (Ph.D. thesis, 1997), which proved that the spherically symmetric 2-spheres of 3-dimensional Schwarzschild space (which is defined to be a totally geodesic, space-like slice of the usual ( 3 + 1 ) (3+1) -dimensional Schwarzschild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.