Deforming a hypersurface by principal radii of curvature and support function

2018/03/22 by Ivaki, Mohammad N. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.08470

Abstract

We study the motion of smooth, closed, strictly convex hypersurfaces in ℝⁿ⁺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 L_p-Christoffel-Minkowski problem φh^1-pσ_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^(1)/(p+k-1) is positive definite, we prove the C^∞ convergence of the normalized flow to a homothetic self-similar solution. One of the highlights of our arguments is that we do not need the constant rank theorem/deformation lemma of Guan-Ma and thus we give a partial answer to a question raised in Guan-Xia. Moreover, for k=n, p≥ n+1, we prove the C^∞ convergence of the normalized flow to a homothetic self-similar solution without imposing any further condition on φ. In the final section of the paper, for 1≤ k

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• The p-th dual Minkowski problem for the k-torsional rigidity corresponding to a k-Hessian equation

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