On Ricci Solitons with Isoparametric Potential Functions

TL;DR

This paper investigates the geometric structure of complete gradient Ricci solitons with isoparametric potential functions, revealing critical level set properties and asymptotic behaviors, advancing understanding of Ricci soliton models.

Contribution

It provides new insights into the structure of Ricci solitons with isoparametric potentials, including critical level set properties and asymptotic analysis, especially for the steady and shrinking cases.

Findings

Critical level set of codimension greater than one in steady Ricci solitons.

Partial results on shrinking Ricci solitons.

Asymptotic behaviors for a specific ansatz.

Abstract

This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with construction of cohomogeneity one models with singular orbits. There is a partial result for the shrinking case. We also study a particular ansatz of popular interest and obtain asymptotic behaviors.