Anisotropic inverse harmonic mean curvature flow

Abstract

We study the evolution of convex hypersurfaces X(·, t) with initial X(’, \mathrm{0})=\theta X_{\mathrm{0}} at a rate equal to H-f along its outer normal, where H is the inverse of harmonic mean curvature of X(’, t), X₀ is a smooth, closed, and uniformly convex hypersurface. We find a {\theta }^{\ast }>\mathrm{0} and a sufficient condition about the anisotropic function f, such that if \theta >{\theta }^{*},∗ , then X(’, t) remains uniformly convex and expands to infinity as t→ +∞ and its scaling, X(’, t)e-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H-log f instead of H-f.

Key words： Curvature flow; parabolic equation; asymptotic behavior

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