Contracting Axially Symmetric Hypersurfaces by Powers of the $$\sigma _k$$-Curvature

Abstract

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in R n+1 and S n+1 by σ α k , where σ k is the k-th elementary symmetric function of the principal curvatures and α ≥ 1/k. We prove that for any n ≥ 3 and any fixed k with 1 ≤ k ≤ n, there exists a constant c(n, k) > 1/k such that that if α lies in the interval [1/k, c(n, k)], then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case 1 < k ≤ n ≤ k 2 , we can choose c(n, k) = 1 k−1 . Our results provide an evidence for the general convergence result without initial curvature pinching conditions. 2010 Mathematics Subject Classification. 53C44, 35B40, 35K55. Key words and phrases. contracting curvature flow, high powers of curvature, k-th elementary symmetric function, axially symmetric hypersurface, curvature flow in sphere.

Non-homothetic convex ancient solutions for flows by high powers of curvature

“…The above identity is proved in [22,Lemma 3.6]; previous similar results can be found in [3] and [25].…”

“…We remark that the statement in [22] requires the hypersurface to have strictly positive curvatures. However, it is easy to see that the result also holds in the weakly convex case.…”

“…However, it is easy to see that the result also holds in the weakly convex case. In fact, the proof in [22] only requires µ > 0 and also works if λ = 0. On the other hand, by (2.8), µ > 0 everywhere on a convex axially symmetric hypersurface except possibly at the poles.…”

“…This time we cannot apply directly Lemma 1, because G is the sum of two terms with different degrees of homogeneity. However, we can adapt the strategy of proof from [22] to our situation and obtain again a useful expression for the gradient terms. Our starting point will be the following identity, which is a consequence of the rotational symmetry and is independent of the homogeneity of F, G: at any stationary point of G with ġ1 = 0 and λ = µ we have, see formula (3.11) in [22],…”

“…Convergence to a round point holds for general speeds [5] if the starting hypersurface satisfies a suitably strong curvature pinching: however, ovaloids do not satisfy any uniform pinching as t → −∞, thus no such property can be expected on the approximants. Even in the rotationally symmetric case, the few available results in homogeneity greater than one [25,22] require some curvature pinching. However, we can exploit a further feature of our approximants, namely that they can be constructed in such a way that the axial curvature is not larger than the other ones at every point.…”