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.

中文翻译：

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余维2中球形曲线的密度缩短曲线的流动。

在本文中，我们通过在密度为\（（M ^ 3_w，\，g_w，\，\ xi）的3维旋转对称空间中以密度为平均曲率流对球形曲线的流动进行了系统的研究。\）密度\（\ xi \）分解为径向部分\（\ varphi \）和角部分\（\ psi \）的总和。我们分析了\（（M ^ 3_w，\，g_w）\）的抛物线性或双曲线性如何限制流向无穷大时的流动行为。