Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

Key Points

• The research aims to establish criteria for stable minimal hypersurfaces in shrinking gradient Ricci solitons.
• Analyzed the properties of smooth stable minimal hypersurfaces in specific Ricci soliton conditions.
• Examined the implications of vanishing second fundamental form and normal Ricci curvature in the context of scalar curvature.
• Demonstrated that compact 2-sided stable minimal hypersurfaces have vanished second fundamental form and normal Ricci curvature.
• Showed that in shrinking gradient Ricci solitons, if an area-minimizing hypersurface exists, the manifold must be splitting.