Deforming a hypersurface by Gauss curvature and support function

Ivaki, Mohammad N.

doi:10.1016/j.jfa.2016.07.003

Cited by 33 publications

(33 citation statements)

References 47 publications

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“…One can see similarly that item (ii) is preserved along the flow. For the case ϕ ≡ 1, k = n, p = −n − 1, preserving a property similar to (ii) played a role in the proofs of [13].…”

Section: An Expanding Flowmentioning

confidence: 97%

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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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“…One can see similarly that item (ii) is preserved along the flow. For the case ϕ ≡ 1, k = n, p = −n − 1, preserving a property similar to (ii) played a role in the proofs of [13].…”

Section: An Expanding Flowmentioning

confidence: 97%

“…For p > 2, ϕ ≡ 1, the C 1 convergence follows from the work of Chow-Gulliver [9] (up-to showing convexity is preserved). For p = −n − 1, k = n, ϕ ≡ 1, the flow was studied in [12,13] and for p > −n − 1, k = n, ϕ ≡ 1 in [3]. Acknowledgment M.I.…”

Section: An Expanding Flowmentioning

confidence: 99%

Deforming a hypersurface by principal radii of curvature and support function

Ivaki

2018

Calc. Var.

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“…In particular, in either case ϕ = 1 or ϕ ≡ 1, −n − 1 < p < 2, we are not aware of any result in the literature on the asymptotic behavior of the flow. The following theorem was proved in [37] regarding the case p = −n − 1, ϕ ≡ 1 (in this case the flow belongs to a family of centro-affine normal flows introduced by Stancu in [60]).…”

Section: Introductionmentioning

confidence: 99%

“…Theorem ( [37]). Let n ≥ 2, p = −n − 1, ϕ ≡ 1 and suppose K 0 has its Santaló point at the origin, i.e., S n u h K0 (u) n+2 dσ(u) = 0.…”

Section: Introductionmentioning

confidence: 99%

A unified flow approach to smooth, even Lp-Minkowski problems

2019

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“…The infimum are in fact achieved in interior points e p (K), see [11]. Moreover, the Jensen, Hölder and Blaschke-Santaló inequality 2 inequalities imply that if V (K) = V (B), then…”

Section: Proof Of Theorem 21mentioning

confidence: 97%