A fully nonlinear locally constrained anisotropic curvature flow

TL;DR

This paper introduces a new fully nonlinear anisotropic curvature flow that preserves convexity and converges to a scaled Wulff shape, providing a novel proof of anisotropic Alexandrov-Fenchel inequalities.

Contribution

It develops a fully nonlinear locally constrained anisotropic curvature flow and proves its smooth exponential convergence to Wulff shapes, offering new insights into anisotropic geometric inequalities.

Findings

Flow exists for all time for smooth convex hypersurfaces.

Flow converges exponentially to a scaled Wulff shape.

Provides a new proof of anisotropic Alexandrov-Fenchel inequalities.

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.