A comparison theorem for steady Ricci solitons

2022/07/09 by Leandro, Benedito, Poveda, Jeferson
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.04259

Abstract

We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton N^n-1×ℝ, where N^n-1 is Ricci flat, or isometric to the Bryant soliton (up to scalings), provided that a couple of geometric conditions inspired by the cigar soliton hold. As an application, we prove that any complete non-compact steady Ricci soliton with positive Ricci curvature controlled by the scalar curvature R, curvature tensor Rm satisfying |Rm|r→ o(1) and R→∞, as r→∞, must be the Bryant soliton. Moreover, we prove that any complete steady soliton with positively pinched Ricci curvature must be Ricci flat.

Related

• Curvature estimates for steady Ricci solitons
• Rigidity of positively curved Steady gradient Ricci solitons on orbifolds
• Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons
• Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature