Isometry theorem of gradient Shrinking Ricci solitons

2020/02/29 by Absos Ali Shaikh, Chandan Kumar Mondal
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear system #Physics #Pure mathematics #Quantum mechanics #Ricci curvature #Ricci flow #Scalar curvature #Soliton #math.DG #msc:53C20 #msc:53C21

paper · pdf · doi:10.1016/j.geomphys.2021.104110

10 pages

arxiv created 2020/05/23 · openalex publication_date 2021/01/13 · arxiv updated 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to \mathbbS⁴ or \mathbbRP⁴ or \mathbbCP². We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.

Citations

• Some Geometry and Analysis on Ricci Solitons
• Three-manifolds with positive Ricci curvature
• Some Nonlinear Problems in Riemannian Geometry
• The formations of singularities in the Ricci Flow
• Ricci solitons: the equation point of view
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products