Abstract:A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons. A similar quantity was found by Feldman, Ilmanen, and Ni [J. Geom. Anal. 15 (2005), pp. 49-62] which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.

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中文翻译：

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Ricci流的稳定长度函数

摘要：Perelman发现Ricci流奇异性的一个基本步骤是发现了单调的体积量，该体积可检测孤子的收缩。Feldman，Ilmanen和Ni发现了相似的数量[J. 几何 肛门 15（2005），第49-62页]中发现了扩展的孤子。当前的工作引入了改进的长度函数，这是迈向稳定孤子单调性公式的第一步。该长度函数以通常的方式生成距离函数，该距离函数显示为满足几个微分不等式，这些微分不等式恰好在满足稳定孤子方程式修改的流形上饱和。

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