Ancient Ricci flows with asymptotic solitons

2021/06/13 by Chan, Pak-Yeung, Ma, Zilu, Zhang, Yongjia
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.06904

Abstract

We study ancient Ricci flows which admit asymptotic solitons in the sense of Perelman. We prove that the asymptotic solitons must coincide with Bamler's tangent flows at infinity. Furthermore, we show that Perelman's ν-functional is uniformly bounded on such ancient solutions; this fact leads to logarithmic Sobolev inequalities and Sobolev inequalities. Lastly, as an important tool for the proofs of the above results, we also show that, for a complete Ricci flow with bounded curvature, the bound of the Nash entropy depends only on the local geometry around an H_n-center of its base point.

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• A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy
• Ancient solutions to the Ricci flow in higher dimensions
• The Soliton-Ricci Flow over Compact Manifolds
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