Ancient Ricci flows with asymptotic solitons

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 $\nu$-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.

References in corpus (3)

• The volume growth of complete gradient shrinking Ricci solitons
• A family of 3d steady gradient solitons that are flying wings
• Perelman-type no breather theorem for noncompact Ricci flows

Cited by in corpus (1)

• On a dichotomy of the curvature decay of steady Ricci soliton