A fully nonlinear locally constrained anisotropic curvature flow

Abstract

Given a smooth positive function F∈C∞(Sn) such that the square of its positive 1-homogeneous extension on Rn+1∖{0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space Rn+1 with F being the support function of the boundary ∂WF. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin the Euclidean space, where Ek denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF, σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in Rn+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.