Abstract
This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the ϕ -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the ϕ -Laplacian operator on closed oriented m -dimensional slant submanifolds in a Sasakian space form M ~ 2 k + 1 ε is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the ϕ -Laplacian on slant submanifold in a sphere S 2 n + 1 with ε = 1 and ϕ = 2 .
Highlights
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
Because there are different boundary conditions on a manifold, one can take a philosophical view of the Dirichlet boundary condition, finding the upper bound for the eigenvalue as a method of investigation for the suitable bound of the Laplacian on the given manifold
Summary
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]. 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. The outcomes of different classes of Riemannian submanifolds indicate that the result of both 1st nonzero eigenvalues depict alike inequalities and have identical upper bounds [20, 22] This result is valid for both Dirichlet and Neumann conditions. The Rayleigh-type variational characterization is observed in first nonzero eigenvalue of Δφ which is given by Λ1,φ, from (cf [30]) This naturally raises the question: Is it possible to generalize the Reilly-type inequalities for submanifolds in spheres through the class almost-contact manifolds which were proved in [1, 20, 21]?
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