The curve shortening flow with density of a spherical curve in codimension two

TL;DR

This paper investigates the evolution of spherical curves under mean curvature flow with density in a 3D rotationally symmetric space, analyzing how the space's geometric properties influence the flow's behavior at infinity.

Contribution

It provides a systematic analysis of the mean curvature flow with density in rotationally symmetric spaces, highlighting the effects of parabolicity and hyperbolicity on the flow's long-term behavior.

Findings

Flow behavior depends on the space's parabolicity or hyperbolicity.

The analysis characterizes conditions for the flow to go to infinity.

The study extends understanding of curvature flows in weighted manifolds.

Abstract

In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flow with density in a 3-dimensional rotationally symmetric space with density $(M^3_w,\:g_w,\:\xi)$ where the density $\xi$ decomposes as sum of a radial part $\varphi$ and an angular part $\psi$. We analyse how either the parabolicity or the hyperbolicity of $(M^3_w,\:g_w)$ condition the behaviour of the flow when the solution goes to infinity.