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

2021/02/15 by Philip Schrader, Schrader, Philip, Glen Wheeler +3 · 1 citation
Mathematics · #34C40 #53E99 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2102.07305

openalex publication_date 2021/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we consider the length functional defined on the space of immersed planar curves. The L²(ds) 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 H¹(ds)-metric. Circles with radius r₀ shrink with r(t) = √W(e^c-2t) under the flow, where W is the Lambert W function and c = r₀² + log r₀². 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.

Citations

• An isoperimetric inequality with applications to curve shortening

Cited by

• Homogeneous Sobolev gradient flow of the length functional

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