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

TL;DR

This paper studies a nonlocal, non-scale-invariant planar geometric flow, revealing unique phenomena like neckpinch singularities, convexity changes, and new convex traveling wave solutions, differing from classical curvature flows.

Contribution

It introduces and analyzes a novel nonlocal geometric flow lacking scale invariance, highlighting its distinct behaviors and solutions compared to classical flows.

Findings

Examples of neckpinch singularity formation

Convexity properties of the flow analyzed

Existence of new convex traveling wave solutions

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.