New quermassintegral and Poincaré type inequalities for non-convex domains

2024/08/12 by Yingxiang Hu, Mohammad N. Ivaki, Hu, Yingxiang +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2408.06057

openalex publication_date 2024/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space ℝⁿ⁺¹, x=(\frac1\fracE_k(κ)E_k-1(κ)-α-⟨ x,ν⟩)ν, k=2,3,…,n-1. Assuming that the initial hypersurface M₀ ⊂ ℝⁿ⁺¹ is star-shaped and its shifted principal curvatures κ=κ+α(1,…,1) lie in the convex set Γ_α,k:=Γ_k-1∩ \λ∈ ℝⁿ: E_k(λ)-αE_k-1(λ)gt;0\, we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincaré type inequality for k-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.

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