Type I singularities in the curve shortening flow associated to a density

TL;DR

This paper investigates Type I singularities in the curve shortening flow with density, analyzing their blow-up behavior and constructing specific examples in curved surfaces with singular densities.

Contribution

It introduces a definition of Type I singularities for the density-affected mean curvature flow and studies their blow-up limits in curved surfaces with singular densities.

Findings

Characterization of Type I singularities in density-affected flow

Construction of curves with only Type I singularities in specific settings

Analysis of blow-up limits for these singularities

Abstract

We define Type I singularities for the mean curvature flow associated to a density $\psi$ ($\psi$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $\psi$-curvature due to the density. We describe a family of curves whose evolution under $\psi$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.