An optimal pinching theorem of minimal Legendrian submanifolds in the unit sphere.

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. Our optimal rigidity results for minimal Legendrian submanifolds in the unit sphere are new and also can be applied to minimal Lagrangian submanifolds in the complex projective space. [ABSTRACT FROM AUTHOR]