The curve shortening problem associated to a density

TL;DR

This paper investigates the evolution of closed curves under a density-weighted mean curvature flow in Euclidean space, focusing on radial densities and convergence to minimal curves.

Contribution

It provides a detailed analysis of the $ ext{psi}$-mean curvature flow for closed curves with radial densities and establishes subconvergence results to $ ext{psi}$-minimal curves.

Findings

Description of the evolution of closed embedded curves under $ ext{psi}$MCF

Conditions for subconvergence to $ ext{psi}$-minimal curves

Analysis of $ ext{psi}$MCF in Euclidean space with radial densities

Abstract

In $\mathbb{R}^n$ with a density $e^\psi$, we study the mean curvature flow associated to the density ($\psi$-mean curvature flow or $\psi$MCF) of a hypersurface. The main results concern with the description of the evolution under $\psi$MCF of a closed embedded curve in the plane with a radial density, and with a statement of subconvergence to a $\psi$-minimal closed curve in a surface under some general circumstances.