Inverse mean curvature flow of rotationally symmetric hypersurfaces

We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in Rn+1 with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as t -> infinity. Our approach is based on a localized version of the parabolic maximum principle. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in Rn+1. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau (Math Z 179:151-168, 1982) and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in R-3.