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Yongjia Zhang

Publications and source records attributed to Yongjia Zhang.

39 records · Page 3Linked to original sources
arXiv2017

Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

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 provides an evidence for Perelman's more general assertion that on a complete ancient solution with nonnegative curvature operator, bounded entropy is equivalent to kappa-noncollapsing.

math.DG↗
arXiv2017

On equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded nonnegative curvature operator, noncollapsing implies bounded entropy. Hence we prove Perelman's assertion under the assumption of bounded geometry on one time slice. In particular, for ancient solutions of dimension three, we need only to assume bounded curvature. We also establish an equality between the asymptotic entropy and the asymptotic reduce volume, which is a result similar to Xu, where he assumes noncollapsing and the Type I curvature bound.

math.DG↗