On Ricci Solitons with Isoparametric Potential Functions

2025/06/04 by Tran, Hung, Yamazaki, Kazuo
#53C25 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53E20 #Secondary 53C22

paper · doi:10.48550/arxiv.2506.03545

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.

Citations

• Geometry and Analysis of Gradient Ricci Solitons in Dimension Four
• Kähler Solitons, Contact Structures, and Isoparametric Functions
• Kähler Gradient Ricci Solitons with Large Symmetry
• SU(2)-invariant steady gradient Ricci solitons on four-manifolds
• Rigidity of four-dimensional Gradient shrinking Ricci solitons
• Eguchi-Hanson singularities in U(2)-invariant Ricci flow
• On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature
• On gradient Ricci solitons with constant scalar curvature
• A Family of Steady Ricci Solitons and Ricci-flat Metrics
• Isoparametric functions on exotic spheres
• Isoparametric functions and exotic spheres
• Isoparametric hypersurfaces with four principal curvatures, II
• Non-Kähler Expanding Ricci Solitons
• On the Completeness of Gradient Ricci Solitons
• Some New Examples of Non-Kähler Ricci Solitons
• On Ricci solitons of cohomogeneity one
• Manifolds with 1/4-pinched Curvature are Space Forms
• Manifolds with positive curvature operators are space forms
• Isoparametric hypersurfaces with four principal curvatures
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Ricci flow with surgery on three-manifolds
• The entropy formula for the Ricci flow and its geometric applications

Related

• On Ricci solitons whose potential is convex
• On gradient Ricci solitons with Symmetry
• Uniqueness of gradient Ricci solitons
• Mild Ricci curvature restrictions for steady gradient Ricci solitons
• Cohomogeneity one shrinking Ricci solitons: an analytic and numerical study