In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

中文翻译：

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具有圆锥奇点的流形的佩雷尔曼熵

在本文中，我们讨论了 Perelman 的 Lambda 泛函、Perelman 的 Ricci 收缩熵以及一类具有孤立圆锥奇点的流形上的 Ricci 扩展熵。在这样的流形上，我们之前的工作存在保留孤立圆锥奇点的奇异 Ricci de Turck 流。我们证明了沿着奇异 Ricci de Turck 流的熵是单调的。我们使用这些熵来表明在奇异设置中，里奇孤子是梯度的，而稳定或扩展的里奇孤子是爱因斯坦。