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

摘要

In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flowwith density in a 3-dimensional rotationally symmetric space with density (M-w(3), gw, 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), gw) conditions the behaviour of the flow when the solution goes to infinity.