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 $(M_w^3,g_w,\xi )$ where the density $\xi$ decomposes as sum of a radial part $\phi$ and an angular part $\psi$. We analyse how either the parabolicity or the hyperbolicity of $(M_w^3,g_w)$ condition the behaviour of the flow when the solution goes to infinity.