In [33], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.

中文翻译：

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Perelman 对古代 Ricci 流的熵

在 [33] 中，第二作者证明了 Perelman 的断言，即对于具有有界和非负曲率算子的古老 Ricci 流，有界熵等效于所有尺度上的非塌陷。在本文中，我们继续讨论。事实证明，曲率算子的非负性不是必要条件，我们只需要假设 Hamilton 迹 Harnack 的结果即可。此外，我们表明该条件适用于具有非负 Ricci 曲率的稳定 Ricci 孤子。