A new characterization of the Calabi torus in the unit sphere

2019/11/19 by Yong Luo, Luo, Yong, Linlin Sun +3
Mathematics · #Characterization (materials science) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Mathematics #Mathematics education #Optics #Physics #Torus #Unit (ring theory) #math.DG

paper · pdf · doi:10.48550/arxiv.1911.08155

arxiv created 2019/11/19 · openalex publication_date 2019/11/19 · arxiv updated 2019/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Legendrian sphere. We also establish an optimal Simons' type integral inequality in terms of the second fundamental form of three dimensional closed minimal Legendrian submanifolds in the unit sphere.

Citations

• Riemannian Geometry of Contact and Symplectic Manifolds
• Local Rigidity Theorems for Minimal Hypersurfaces
• On totally real submanifolds
• Rigidity of compact minimal submanifolds in a unit sphere
• An intrinsic rigidity theorem for minimal submanifolds in a sphere
• Minimal Varieties in Riemannian Manifolds
• Submanifolds with Constant Mean Curvature
• Rigidity of closed CSL submanifolds in the unit sphere

Related

• The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
• On conformally flat minimal Legendrian submanifolds in the unit sphere
• Rigidity of closed CSL submanifolds in the unit sphere
• On Yau rigidity theorem for minimal submanifolds in spheres
• Some minimal submanifolds generalizing the Clifford torus