Asymptotic convergence for a class of inverse mean curvature flows in ℝⁿ⁺¹

Abstract

We consider star-shaped, strictly mean convex and closed hypersurfaces expanding by a class of inverse mean curvature flows in R n + 1 \mathbb {R}^{n+1} , and we prove that this evolution exists for all time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling.

A class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold

“…It is obvious thatξ ξ near (y 0 , t 0 ) and they are equal at this point. At the same time, we can show [2,Theorem 4.2]). Now we define a function w = log(h 1 1 ) + log χ + log Θ near the point (y 0 , t 0 ), such that it attains its maximum w max at (y 0 , t 0 ) and w max = u(y 0 , t 0 ).…”

“…They showed there exists a uniformly convex solution for all time and the hypersurfaces converge to a round point when α k + 1. Chen et al [2] considered the inverse mean curvature analogue of (1.1) in…”

An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$

“…Remark 1.1. (1) This work was firstly announced by us in a previous work [5], which actually corresponds to the higher dimensional case of the work here. Besides, we also mentioned this work in series works [3,6] later.…”

“…Denote by u ξ := D ∂ ξ u, and u ξξ := D ∂ ξ D ∂ ξ u the covariant derivatives of u w.r.t. the metric g H 1 (1) , where D is the covariant connection on H 1 (1). Let ∇ be the Levi-Civita connection of M t w.r.t.…”

“…One might find that our previous works on inverse curvature flows (see, e.g.,[1,3,4,5,6,9]) have used…”