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

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.