In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results in [15, 16, 29] to a wider class of curvature flows.

中文翻译：

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自膨胀器通过齐次函数逆曲率流动

在本文中，我们通过欧几里得空间中主曲率的齐次对称函数，研究了一大类抛物线逆曲率流的自展开解。这些流包括平均曲率逆流和文献中的许多非线性流。我们首先表明，对于任何这些流动，唯一紧凑的自膨胀机是圆形球体。其次，我们表明，对于具有渐近圆柱端的所有这些流，完全非紧凑的自展开器必须是旋转对称的。第三，我们表明，当这种流动是均匀抛物线形时，存在完整的旋转对称自扩张器，它们渐近于两个半径不同的圆柱体。这些将[15、16、29]中的一些早期结果扩展到更广泛的曲率流类别。