A fully nonlinear locally constrained anisotropic curvature flow. (April 2022) - Wei, Yong; Xiong, Changwei
Abstract: Given a smooth positive function F ∈ C ∞ ( S n ) such that the square of its positive 1-homogeneous extension on R n + 1 ∖ { 0 } is uniformly convex, the Wulff shape W F is a smooth uniformly convex body in the Euclidean space R n + 1 with F being the support function of the boundary ∂ W F . In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂ ∂ t X = ( 1 − E k 1 / k σ F ) ν F, k = 2, …, n in the Euclidean space, where E k denotes the normalized k th anisotropic mean curvature with respect to the Wulff shape W F, σ F the anisotropic support function and ν F the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in R n + 1 ( n ≥ 2 ), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov–Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space.