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The curve shortening flow with density of a spherical curve in codimension two. (English) Zbl 1473.53106

Summary: 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 \((M^3_w,\,g_w)\) conditions the behaviour of the flow when the solution goes to infinity.

MSC:

53E10 Flows related to mean curvature
35R01 PDEs on manifolds
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