In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota’s work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies $\kappa$‑noncollapsing on all scales. This result is used by the author [21] to prove Perelman’s assertion that on an ancient solution to the Ricci flow with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales; see section 11 in [17].

中文翻译：

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里氏流的古代解的熵，无塌陷和间隙定理

在本文中，我们讨论了Ricci流的古代解的渐近熵。我们证明了古代解的一个缺口定理，该定理可以看作是横田著作的一个熵对应物。另外，我们证明，在具有非负曲率算子的完整古代解的一个时间片上的某些假设下，有限渐近熵意味着在所有尺度上$ \ kappa $-不塌陷。作者[21]使用该结果证明了佩雷尔曼的断言，即在有界非负曲率算子的Ricci流的古老解决方案中，有界熵等效于所有尺度上的无塌陷；请参阅[17]中的第11节。