On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance

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Abstract

We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow.
In particular, we give examples of neckpinch singularity formation, and we discuss convexity properties of the evolution.
We also take into account traveling waves for this geometric flow, showing that a new family of C 1 , 1 and convex traveling sets arises in this setting.
Original languageEnglish
Pages (from-to)1-21
JournalJournal of the London Mathematical Society
DOIs
Publication statusE-pub ahead of print - 20 Jul 2018

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Geometric Flows
Mean Curvature Flow
Scale Invariance
Curve
Traveling Wave
Convexity
Fractional
Curvature
Scaling
Singularity
Invariant
Family

Cite this

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title = "On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance",
abstract = "We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow.In particular, we give examples of neckpinch singularity formation, and we discuss convexity properties of the evolution.We also take into account traveling waves for this geometric flow, showing that a new family of C 1 , 1 and convex traveling sets arises in this setting.",
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AB - We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow.In particular, we give examples of neckpinch singularity formation, and we discuss convexity properties of the evolution.We also take into account traveling waves for this geometric flow, showing that a new family of C 1 , 1 and convex traveling sets arises in this setting.

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