**Papers Published**

- Bray, H; Morgan, F,
*An isoperimetric comparison theorem for schwarzschild space and other manifolds*, Proceedings of the American Mathematical Society, vol. 130 no. 5 (January, 2002), pp. 1467-1472

(last updated on 2022/08/16)**Abstract:**

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric (n - 1)-spheres of a spherically symmetric n-manifold are isoperimetric hypersurfaces, meaning that they minimize (n - 1)-dimensional area among hypersurfaces enclosing the same 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)-dimensional Schwarzsehild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.