2021/12/21 by Pak-Yeung Chan, Chan, Pak-Yeung, Zilu Ma +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2112.11025
openalex publication_date 2021/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first show that any 4-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy |Rm|≤ cR for some positive constant c. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for 4-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a 3-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Chodosh 14 in dimension three and show that any 3-dimensional gradient Ricci expander C2 asymptotic to (C(\mathbb S2), dt2+αt2 g_\mathbbS2) is rotationally symmetric, where α∈ (0,1] is a constant and g_\mathbbS2 is the standard metric on \mathbbS2 with constant curvature 1.