A fully nonlinear locally constrained anisotropic curvature flow

Yong Wei; Changwei Xiong

Paper · Mar 30, 2021

A fully nonlinear locally constrained anisotropic curvature flow

Yong Wei, Changwei Xiong

math.DGWider flowsarXiv:2103.16088

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Abstract

Given a smooth positive function $F \in C^{\infty} (S^{n})$ 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 $\partial W_{F}$ . In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow \beginequation* \frac{\partial }{\partial t}X=(1-E_k^1/kσ_F)ν_F,\quad k=2,\cdots,n \endequation* 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 $R^{n + 1}$ ( $n \geq 2$ ), 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.

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Submitted: Mar 30, 2021
Subjects: math.DG, math.AP
Journal: Nonlinear Analysis, vol. 217, April 2022, 112760
DOI: 10.1016/j.na.2021.112760

Comments: 30 pages

Other curvature flows Volume-preserving & constrained flows Anisotropic & nonlocal flows Homogeneous spaces Long-time existence & convergence Geometric inequalities

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.

[1] 11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.

A fully nonlinear locally constrained anisotropic curvature flow

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