A fully nonlinear locally constrained anisotropic curvature flow

Abstract

Given a smooth positive function $F\in C^{\infty}(\mathbb{S}^n)$ such that the square of its positive $1$-homogeneous extension on $\mathbb{R}^{n+1}\setminus \{0\}$ is uniformly convex, the Wulff shape $W_F$ is a smooth uniformly convex body in the Euclidean space $\mathbb{R}^{n+1}$ with $F$ being the support function of the boundary $\partial W_F$. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow \begin{equation*} \frac{\partial }{\partial t}X=(1-E_k^{1/k}\sigma_F)\nu_F,\quad k=2,\cdots,n \end{equation*} in the Euclidean space, where $E_k$ denotes the normalized $k$th anisotropic mean curvature with respect to the Wulff shape $W_F$, $\sigma_F$ the anisotropic support function and $\nu_F$ the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in $\mathbb{R}^{n+1}$ ($n\geq 2$), 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.