Isoperimetric estimate for the Ricci flow on \(S^2\times S^1\)
The author studies the dilation limit of \(S^2\times S^1\) with a warped product metric under the Ricci flow. He first proves that the isoperimetric ratio on the base manifold \(S^2\) has a lower bound. This fact excludes the case of \(\Sigma ^2\times \mathbb{R}\) (where \(\Sigma ^2\) is the cigar soliton on the surface) as the dilation limit. It confirms the conjecture by \textit{R. S. Hamilton} in this particular case [see ``The formation of singularities in the Ricci flow'', Proceedings of the conference on geometry and topology held at Harvard University, Cambridge, MA, USA, April 23--25, 1993. Cambridge, MA: International Press. Surv. Differ. Geom., Suppl. J. Differ. Geom. 2, 7--136 (1995; Zbl 0867.53030)]. The author also proves a monotonicity result on the isoperimetric ratio.
scientific article; zbMATH DE number 5015982