On steady and expanding Ricci solitons with asymptotic symmetries

TL;DR

This paper proves symmetry principles and uniqueness results for certain steady and expanding Ricci solitons with specific asymptotic geometries and symmetries.

Contribution

It introduces a symmetry principle for asymptotically cylindrical and conical Ricci solitons and characterizes Bryant solitons uniquely under these conditions.

Findings

Bryant steady soliton is unique with a round spherical link and rigidity condition.

Similar uniqueness result for Bryant's expanding solitons.

Global symmetry established for GRSs with quotient-Berger sphere asymptotics.

Abstract

We establish a symmetry principle for asymptotically cylindrical steady gradient Ricci solitons (GRSs) and asymptotically conical expanding GRSs with homogeneous links. Using this, we show that the Bryant steady soliton is the unique asymptotically cylindrical steady GRS that has a round spherical link and satisfies a particular quantitative rigidity condition. A similar characterization is proved for Bryant's expanding solitons. Finally, we establish a global symmetry result for GRSs which exhibit the aforementioned asymptotics with quotient-Berger sphere asymptotic links.