The isoperimetric inequality in steady Ricci solitons

TL;DR

This paper proves the isoperimetric inequality holds in specific steady Ricci solitons, namely the cigar and Bryant solitons, using warped product metrics and existing geometric results.

Contribution

It establishes the validity of the isoperimetric inequality in certain steady Ricci solitons, extending geometric understanding of these manifolds.

Findings

Isoperimetric inequality holds in cigar steady soliton

Isoperimetric inequality holds in Bryant steady soliton

Utilizes Guan-Li-Wang result for warped product metrics

Abstract

We prove that the isoperimetric inequality is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, we utilize the result of Guan-Li-Wang to get our conclusion. For the sake of the soliton structure, we believe that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally satisfied for steady Ricci solitons.