Abstract
In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.
Highlights
In the last few years, there has been attentioned to the classification of Lagrangian submanifolds
Lagrangian submanifolds give an impression being of foliations in the cotangent bundle, and Hamilton-Jacobi type leads to the classification via partial differential equation
The Ricci tensor is involving in the curvature spacetime, which finds the degree where matter will incline to converge or diverge in time
Summary
In the last few years, there has been attentioned to the classification of Lagrangian submanifolds. By considered that compact submanifold Mn immersed in the Euclidean sphere Sn+p or Euclidean space Rn+p, Jiancheng and Zhang [29] derived the Simons-type [30] inequalities about the first eigenvalue μ1 and the squared norm of the second fundamental form S without using the condition that submanifold M is minimal They established a lower bound for S if it is constant. By using the techniques of conformal vector field which have prominent appearance in deriving characterizations of spaces and have high-level geometry in the theory of relativity and mechanics, Deshmukh and AlSolamy [32] proved that an n-dimensional compact connected Riemannian manifold whose Ricci curvature satisfied the bound 0 < Ric ≤ ðn − 1Þð2 − nc/μ1Þc for a constant c and μ1 is the first nonzero eigenvalue of the Laplace operator; Mn is isometric to SnðcÞ if Mn admitted a nonzero conformal gradient vector field.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
