First eigenvalue of the $p$-Laplacian on Kähler manifolds

Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms

“…On compact submanifold, they used the Wentzel-Laplace operator having boundary in Euclidean space. Following the same pattern, for Neumann and Dirichlet boundary restrictions, Blacker and Seto [3] evidenced the Lichnerowicz-type lower bound for the first nonzero eigenvalue of the ϕ-Laplacian. They used the Hessian decomposition on Kaehler manifolds having a positive Ricci curvature.…”

“…Finding the bound of the eigenvalue for the Laplacian on a given manifold is a key aspect in Riemannian geometry, and there are different classes of submanifolds such as slant submanifolds, CR-submanifolds, and singular submanifolds, which motivates further exploration and attracts many researchers from different research areas [1][2][3][4][5][6][7][8][9][10][11]. A major objective is to study the eigenvalue that appears as solutions of the Dirichlet or Neumann boundary value problems for curvature functions.…”

“…where the volume element of N m is denoted by dV. It can be seen in literature that many authors prompted to work in such inequalities for different ambient spaces after the breakthrough of inequality (3). In Minkowski spaces, the upper bound for Finsler submanifold is proposed by both Zeng and He [14].…”

About Bounds for Eigenvalues of the Laplacian with Density

“…(see, e.g., [2, equation (3)]). We recall that our sign convention for the operators ∆ and ∆ ∂ is the opposite of that in [2]. Moreover,…”

“…The particular cases where α = 1 and α = n−2 n correspond to the problems mentioned above, whose eigenvalues are respectively given by (2) and (3).…”