On the H¹(ds)-gradient flow for the length functional

Philip Schrader; Glen Wheeler; Valentina-Mira Wheeler

Paper · Feb 15, 2021

On the $H^{1} (d s)$ -gradient flow for the length functional

Philip Schrader, Glen Wheeler, Valentina-Mira Wheeler

math.DGWider flowsarXiv:2102.07305v2

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Abstract

In this article we consider the length functional defined on the space of immersed planar curves. The $L^{2} (d s)$ Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the triviality of the metric topology in this space, we consider the gradient flow of the length functional with respect to the $H^{1} (d s)$ -metric. Circles with radius $r_{0}$ shrink with $r (t) = W (e^{c - 2 t})$ under the flow, where $W$ is the Lambert $W$ function and $c = r_{0}^{2} + lo g r_{0}^{2}$ . We conduct a thorough study of this flow, giving existence of eternal solutions and convergence for general initial data, preservation of regularity in various spaces, qualitative properties of the flow after an appropriate rescaling, and numerical simulations.

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Submitted: Feb 15, 2021
Last revised: Mar 3, 2021 (v2)
Subjects: math.DG, math.AP

Comments: 37 pages; corrected and added some remarks, notation, typos

Curve shortening Ancient solutions Numerics & visualization

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Background reading

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

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

6D.11
Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.
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.
12C.4
Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof
Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.

[1] 6D.11
Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

[2] 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.

[3] 12C.4
Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof
Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.

On the $H^{1} (d s)$ -gradient flow for the length functional

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On the H1(ds)-gradient flow for the length functional

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On the $H^{1} (d s)$ -gradient flow for the length functional