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

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 (M3 w, gw,ξ) where the density ξ decomposes as sum of a radial part ϕ and an angular part ψ. We analyse how either the parabolicity or the hyperbolicity of (M3 w, gw) conditions the behaviour of the flow when the solution goes to infinity.