Paper · Mar 9, 2024

Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

Marco Cicalese, Andrea Kubin

math.APWider flowsarXiv:2403.05947
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Abstract

We investigate the area-preserving mean-curvature-type motion of a two-dimensional lattice crystal obtained by coupling constrained minimizing movements scheme introduced by Almgren, Taylor and Wang with a discrete-to-continuous analysis. We first examine the continuum counterpart of the model and establish the existence and uniqueness of the flat flow, originating from a rectangle. Additionally, we characterize the governing system of ordinary differential equations. Subsequently, in the atomistic setting, we identify geometric properties of the discrete-in-time flow and describe the governing system of finite-difference inclusions. Finally, in the limit where both spatial and time scales vanish at the same rate, we prove that a discrete-to-continuum evolution is expressed through a system of differential inclusions which does never reduce to a system of ODEs.

Mean curvature flowVolume-preserving & constrained flowsSurfacesWeak & singular flowsStability & uniqueness

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    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Andrea Kubin · Shared terms “movement scheme”, “minimizing movements”

  2. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

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    Topics Volume-preserving & constrained flows, Stability & uniqueness · Same author Andrea Kubin

  3. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”, “movement scheme”, “minimizing movements”

  4. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”, “movement scheme”, “minimizing movements”

  5. De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

    Tim Laux and Andrea PoiattiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”, “movement scheme”, “Taylor”

  6. Weak-strong uniqueness for volume-preserving mean curvature flow

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    Topics Volume-preserving & constrained flows, Weak & singular flows, Stability & uniqueness

  7. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”, “movement scheme”, “minimizing movements”

  8. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “area preserving”, “flat flow”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.