Deforming a hypersurface by principal radii of curvature and support function

Abstract

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

A class of anisotropic expanding curvature flows

“…We have the same result for k = n without any constraint on positive smooth function f , which recovers the result of Chou and Wang [11]. When β = 1, the flow has been studied by Ivaki recently [24]. In this case the self-similar solution of the flow ∂ t u = f u α σ β k is the solution to f u α−1 σ k = c which is just the L p Christoffel-Minkowski problem for p = 2 − α.…”

“…Remark 1.1. When β = 1, our second flow (1.1) is just the one that Ivaki has studied recently [24]. In [24], Ivaki employed the functional in [20] to prove that the flow ∂ t u = f u α σ k has a unique smooth solution, and the rescaled flow converges smoothly to a homothetic self-similar solution which is a solution f u α−1 σ k = c for k < n, α ≤ 1 − k and positive function f satisfies that…”

“…gave the proofs for a class of L p Christoffel-Minkowski problems. On the other hand, as a nature extension, anisotropic flows usually provide alternative proofs and smooth category approach of the existence of solutions to elliptic PDEs arising in convex body geometry, see [30,10,19,26,24] etc.. In this paper, we consider two expanding flows of the convex hypersurfaces at the speeds of u α σ β k (λ 1 , ...λ n ) and f u α σ β k (λ 1 , ...λ n ) respectively, where u is the support function, f is a smooth positive function on S n , α, β ∈ R 1 , β > 1 k and σ k (λ 1 , ..., λ n ) is the k-th symmetric polynomial of the principal curvature radii of the hypersurface, k is an integer and 1 ≤ k ≤ n. Generally, for the flow with the speed of high powers of curvatures, it is required that the initial hypersurface is uniformly convex and satisfies a suitable pinching conditions, so as to preserve the uniformly convexity and converge to a sphere ( [1], [6]).…”

“…It has been obtained by Chou and Wang in [11]. Before Ivaki [24], Hu et al [21] proved the existence result for the L p Christoffel Minkowski problem for p = 2 − α: f u α−1 σ k = 1, α ≤ 1 − k, 1 ≤ k ≤ n−1 for any positive function f satisfying…”

Deforming a Convex Hypersurface by Anisotropic Curvature Flows

“…Hu, Ma & Shen in [17] proved the existence of convex solutions to the L p -Christoffel-Minkowski problem for p ≥ k + 1 under appropriate conditions. Using the methods of geometric flows, Ivaki in [21] and then Sheng & Yi in [33] also gave the existence of smooth convex solutions to the L p -Christoffel-Minkowski problem for p ≥ k + 1. In case 1 < p < k + 1, Guan & Xia in [13] established the existence of convex body with prescribed k-th even p-area measures.…”