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In [ZY2], the second author proved Perelman's assertion, namely, for an ancient solution to the 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.

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This note is a continuation of [7]. We shall show that an ancient Ricci flow with uniformly bounded Nash entropy also has uniformly bounded $\nu $-functional. Consequently, on such an ancient solution, there are uniform logarithmic Sobolev and Sobolev inequalities. We emphasize that the main theorem in this paper is true so long as the theory in [3] is valid, and in particular, when the underlying manifold is closed.

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