Paper · Mar 10, 2019

Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

Yoshiki Jikumaru, Miyuki Koiso

math.DGWider flowsarXiv:1903.03958
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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.

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Submitted
Mar 10, 2019
Subjects
math.DG

Comments: 19 pages, 7 figures. arXiv admin note: text overlap with arXiv:1903.03951

Mean curvature flowAnisotropic & nonlocal flows3-manifoldsWeak & singular flowsStability & uniqueness

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  1. Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “Wulff shape”

  2. Motion of level sets by inverse anisotropic mean curvature

    Francesco Della Pietra, Nunzia Gavitone and Chao XiaWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness

  3. Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

    Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows

  4. Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions

    Eric Kim and Dohyun KwonWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “Wulff shape”, “isotropic”

  5. Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

    Annalisa Cesaroni, Heiko Kröner and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Shared terms “isotropic”, “rescaling”

  6. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness · Shared terms “anisotropic energy”

  7. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows

  8. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness

Read around it

Background reading

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

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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