Asymptotic behavior of flows by powers of the Gaussian curvature

Brendle, Simon; Choi, Kyeongsu; Daskalopoulos, Panagiota

doi:10.4310/acta.2017.v219.n1.a1

Cited by 144 publications

(118 citation statements)

References 19 publications

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“…That is, for some ε > 0 and an open neighborhood O uτ of u τ , we have ξ(u, t) ≤ (hλ 1 )(u, t) for all (u, t) ∈ O uτ × (τ − ε, τ ] and ξ(u τ , τ ) = (hλ 1 )(u τ , τ ). With similar calculations as in [6,Lemma 5] at (u τ , τ ) we obtain…”

Section: Expand This Last Expression Formentioning

confidence: 62%

Deforming a hypersurface by principal radii of curvature and support function

Ivaki

2018

Calc. Var.

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We study the motion of smooth, closed, strictly convex hypersurfaces in R n+1 expanding in the direction of their normal vector field with speed depending on the kth elementary symmetric polynomial of the principal radii of curvature σ k and support function h. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known Lp-Christoffel-Minkowski problem ϕh 1−p σ k = c. Here ϕ is a preassigned positive smooth function defined on the unit sphere, and c is a positive constant. For 1 ≤ k ≤ n − 1, p ≥ k + 1, assuming the spherical hessian of ϕ where p ∈ R, E i is the ith elementary symmetric polynomial of principal curvatures for 0 ≤ i ≤ n normalized so that E i (1, . . . , 1) = 1 (and E 0 ≡ 1), ν(·, t) is the outer unit normal vector of M t := F (M n , t) and ϕ is a positive smooth function defined on the unit sphere S n .Assuming M t is strictly convex, its support function as a function on the unit sphere is given byWriteḡ and∇ for the standard round metric and the Levi-Civita connection of S n . Recall that the principal radii of curvature are the eigenvalues of the matrix r ij :=∇ i∇j h +ḡ ij h

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Section: Expand This Last Expression Formentioning

confidence: 62%

Deforming a hypersurface by principal radii of curvature and support function

Ivaki

2018

Calc. Var.

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“…When ǫ = 0 and k = 2, the flow (1.1) corresponds to the flow by powers by the scalar curvature, which was studied by Alessandroni and Sinestrari in [1]. When ǫ = 0 and k = n, the flow (1.1) is the flow by powers of the Gauss curvature, which has been well studied, we refer to [4,5,6,11,12,14,17,27] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

2020

J Geom Anal

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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.

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“…The proof is a delicate application of the maximum principle to two test functions W = F κ1 − β−1 β Φ and Z, where κ 1 is the smallest principal curvature of M . The idea comes from [6,5] and is used in [9]. The following lemma is employed to analyze the maximum points of W .…”

Section: Analysis At the Maximum Points Of Wmentioning

confidence: 99%

“…Self-similar solutions are important in the study of mean curvature flow and powers of Gauss curvature flow in Euclidean space, since they describe the asymptotic behaviors near the singularities (See [11,7,4,10] etc). Remarkable results due to Huisken [11] and Choi-Daskalopoulos [6], Brendle-Choi-Daskalopoulos [5] show the uniqueness of closed self-similar solutions for mean curvature flows and powers of Gauss curvature flows respectively. Although relation between self-similar solutions of general curvature flows and their singularities is unclear now, there have been some study on rigidity of closed self-similar solutions of curvature flows, for instance, [12], [9], etc.…”

Section: Introductionmentioning

confidence: 99%

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Self-similar solutions of curvature flows in warped products

Gao

2019

Differential Geometry and its Applications

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In this paper we study self-similar solutions in warped products satisfying F − F =ḡ(λ(r)∂r , ν), where F is a nonnegative constant and F is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such F . We also obtain a similar uniqueness result in hyperbolic space H 3 for Gauss curvature F and F ≥ 1.2010 Mathematics Subject Classification. 53C44, 53C40.

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Asymptotic behavior of flows by powers of the Gaussian curvature

Cited by 144 publications

References 19 publications

Deforming a hypersurface by principal radii of curvature and support function

Deforming a hypersurface by principal radii of curvature and support function

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

Self-similar solutions of curvature flows in warped products

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