Paper · May 23, 2019

On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

Harald Garcke, Bogdan-Vasile Matioc

math.APWider flowsarXiv:1905.09675
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Abstract

We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing the evolution of the points of the evolving surface that lie on the rotation axis. For the fully nonlinear and degenerate parabolic problem we establish the well-posedness property in the setting of classical solutions. Besides we prove that the problem features the effect of parabolic smoothing.

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Submitted
May 23, 2019
Subjects
math.AP
Journal
J. Evol. Equ., 21:201-224, 2021

Comments: 19 pages

Mean curvature flowRotational symmetry & warped products

Its neighbourhood

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  1. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”, “formulated”

  2. Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds

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    Topics Rotational symmetry & warped products · Same author Harald Garcke · Shared terms “rotationally symmetric”

  3. Uniqueness of two-convex closed ancient solutions to the mean curvature flow

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    Topics Rotational symmetry & warped products · Shared terms “symmetric closed”, “rotationally symmetric”

  4. Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

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    Topics Mean curvature flow · Same author Harald Garcke · Shared terms “evolution equation”, “consisting”

  5. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows

    Topics Mean curvature flow · Same author Harald Garcke · Shared terms “formulated”, “evolution equation”, “consisting”

  6. Rotational symmetry of solutions of mean curvature flow coming out of a double cone

    Letian ChenWider flows

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  7. Type-II singularities of two-convex immersed mean curvature flow

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    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  8. Uniqueness of entire graphs evolving by Mean Curvature flow

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    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.