Paper · Aug 12, 2024
New quermassintegral and Poincaré type inequalities for non-convex domains
As on arXiv
Abstract
In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space , \beginalign* \dotx=\left(\frac1{\frac{E_k(\hatκ)}{E_k-1(\hatκ)}-α}-\langle x,ν\rangle\right)ν, \quad k=2,3,\ldots,n-1. \endalign* Assuming that the initial hypersurface is star-shaped and its shifted principal curvatures lie in the convex set \beginalign* Γ_α,k:=Γ_k-1\cap \{λ\in \mathbbR^n:\, E_k(λ)-αE_k-1(λ)>0\}, \endalign* 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 -convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.
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