[Paper Review] Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

Yoshiki Jikumaru, Miyuki Koiso|arXiv (Cornell University)|Mar 10, 2019

Geometric Analysis and Curvature Flows17 references4 citations

TL;DR

This paper demonstrates the non-uniqueness of closed embedded hypersurfaces with constant anisotropic mean curvature (CAMC) in ℝⁿ⁺¹, even for genus-zero surfaces in ℝ³, by constructing explicit C² anisotropy functions γ that are not convex. Unlike the isotropic case, where the Wulff shape is the unique minimizer, the authors show that non-convex γ can yield CAMC hypersurfaces not homothetic to the Wulff shape, and further construct nontrivial self-similar shrinking solutions for anisotropic mean curvature flow.

ABSTRACT

An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. We prove that, unlike the isotropic case, there exists an anisotropic energy density function such that there exist closed embedded equilibrium surfaces with genus zero in the three-dimensional Euclidean space each of which is not (any homothety and translation of) the Wulff shape. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. These results are generalized to hypersurfaces in the (n+1)-dimensional Euclidean space.

Motivation & Objective

• To investigate whether closed embedded CAMC hypersurfaces in ℝⁿ⁺¹ must be homothetic to the Wulff shape, especially under topological or stability constraints.
• To challenge the expectation that CAMC hypersurfaces are unique up to homothety and translation when the Wulff shape is not smooth or convex.
• To construct explicit examples of non-convex anisotropy functions γ for which there exist closed embedded CAMC hypersurfaces not homothetic to the Wulff shape.
• To extend these results to the anisotropic mean curvature flow by constructing nontrivial self-similar shrinking solutions.

Proposed method

• Constructing a C², non-convex anisotropy function γ:Sⁿ→ℝ>₀ using spherical harmonics and rotational symmetry.
• Defining the Cahn-Hoffman map ξγ(ν) = Dγ|ν + γ(ν)ν to characterize the Wulff shape as the image of γ under this map.
• Using the Cahn-Hoffman field to define a piecewise-C² weak immersion X:M→ℝⁿ⁺¹ whose anisotropic mean curvature is constant.
• Proving that the image of the Cahn-Hoffman map for γ contains closed embedded surfaces with constant anisotropic mean curvature that are not homothetic to the Wulff shape.
• Verifying that the anisotropic mean curvature Λ is constant on these surfaces by direct computation using the formula Λ = (1/n)(−divₘDγ + nHγ).
• Applying the same construction to the anisotropic mean curvature flow by showing that certain surfaces evolve self-similarly under the flow ∂Xₜ/∂t = Λₜξₜ.

Experimental results

Research questions

• RQ1 Can there exist closed embedded CAMC hypersurfaces in ℝⁿ⁺¹ that are not homothetic to the Wulff shape for a non-convex anisotropy function γ?
• RQ2 Is the uniqueness of CAMC hypersurfaces up to homothety and translation preserved when γ is not convex, even for genus-zero surfaces in ℝ³?
• RQ3 Do non-convex anisotropy functions give rise to nontrivial self-similar shrinking solutions of the anisotropic mean curvature flow?
• RQ4 Can the Cahn-Hoffman map be used to construct explicit examples of non-Wulff CAMC surfaces?
• RQ5 What conditions on γ ensure that all CAMC hypersurfaces are homothetic to the Wulff shape, and where does this fail?

Key findings

• The paper constructs a C∞ function γ:S¹→ℝ>₀ that is not convex and yields a closed embedded CAMC curve in ℝ² not homothetic to the Wulff shape Wγ.
• For n=2, a C∞ non-convex γ:S²→ℝ>₀ is constructed such that there exist closed embedded CAMC surfaces in ℝ³ with genus zero that are not homothetic to Wγ.
• The constructed examples are subsets of the image of the Cahn-Hoffman map ξγ, and their anisotropic mean curvature is constant and equal to -1.
• The Wulff shape Wγ₂ for the constructed γ₂ is not a smooth surface and differs from the embedded CAMC surfaces, which are piecewise-C∞ and not homothetic to Wγ₂.
• The paper provides explicit formulas for γ₂ on S² and its Cahn-Hoffman map ξγ₂, showing that the image contains multiple distinct CAMC surfaces.
• Nontrivial self-similar shrinking solutions of the anisotropic mean curvature flow are constructed, with ∂Xₜ/∂t = Λₜξₜ and Λₜ = -1/√(2(c−t)), demonstrating non-uniqueness in the flow context.