2024/03/03 by Peiyi Wu, Ling Yang, Wu, Pei-Yi +1
Engineering · Mathematics · #53C24 #53C42 #Differential Geometry (math.DG) #Elasticity and Wave Propagation #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2403.01453
openalex publication_date 2024/03/03 · openalex created_date 2024/03/07 · openalex updated_date 2026/07/28
In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu's inequality on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.