Deforming Pinched Hypersurfaces of the Hyperbolic Space by Powers of the Mean Curvature Into Spheres

Abstract

Abstract. This paper concerns closed hypersurfaces of dimension n ≥ 2 in the hyperbolic space H n+1 κ of constant sectional curvature κ evolving in direction of its normal vector, where the speed equals a power β ≥ 1 of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and β, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in H n+1 κ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of H n+1 κ .

Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

“…Similar results were obtained for powers of the scalar curvature [1,2], and for K β with β > 1/(n + 2) without pinching condition [14]. Recently, with different pinching condition, Schulze [38,39], Guo, Li and Wu [24,25] proved that the convergence of flows with speeds equal to a power of the mean curvature were still hold in Euclidean space and in hyperbolic space, respectively.…”

Contracting convex hypersurfaces by functions of the mean curvature

“…In the previous paper [5], the author, together with Li and Wu, extended Schulzes results to h-convex hypersurfaces in the hyperbolic space, and showed that if the initial hypersurface has mean curvature bounded below, the positive power mean curvature flow has a unique, smooth solution on a finite time interval, and converges to a point if the initial hypersurface is strictly h-convex for the case that 0 < β < 1, or weakly convex for β ≥ 1. Moreover, for the H β -flow case with β ≥ 1, it has been found that if the initial hypersurface M 0 has the ratio of largest to smallest principal curvatures close enough to 1 at every point, then the evolving hypersurfaces contract to a round point: This was first shown in the Euclidean space setting by Schulze [21], then in the hyperbolic space setting by the author, Li and Wu [6].…”

Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

“…To control the pinching of the principal curvatures along the flow (1.1) of Euclidean spaces, Schulze, in [38], following an idea of Tso [42], looked at a test function K/H n , which was also considered in [12]. Furthmore two analogous quantities K/ Hn and K/F n were taken into consideration in [18,19] and [34] respectively. In this section, we use test functions Q 1 = K/H n in the case of concave F and Q 2 = K/F n in the case of convex F respectively.…”

“…So the next problem we meet is: what is the suitable geometric quantity for the mixed volume preserving F β -flow. Meanwhile we observe that most of the literature in the investigation of evolution equations, requires an additional assumption of a suitable geometric quantity on the initial hypersurface for example, of K/H n used for the gauss curvature flow [13], the power mean curvature flow [37,18] and the volume preserving [12,19], R/H 2 used for the flow by the square root of the scalar curvature [14], and K/F n used for the mixed volume preserving F -flow [34]. We realized that the two scalars K/F n for convex F and K/H n for concave F should be good candidates for our flow.…”

“…Inspired by these results of [12,18,34], in this paper we replace the speed function F (λ) in [34] by a power of the F (λ), namely we study the long time existence and the convergence of the flow (1.1) with the speed Φ given by a power of the curvature function, namely…”