Anisotropic inverse harmonic mean curvature flow

Abstract

We study the evolution of convex hypersurfaces Open image in new window with initial Open image in new window at a rate equal to H — f along its outer normal, where H is the inverse of harmonic mean curvature of Open image in new window is a smooth, closed, and uniformly convex hypersurface. We find a θ* > 0 and a sufficient condition about the anisotropic function f, such that if θ > θ*, then Open image in new window remains uniformly convex and expands to infinity as t → + ∞ and its scaling, Open image in new window, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is logH-log f instead of H-f.