Eigenvalues for the Clamped Plate Problem of \(\mathfrak{L}_{\nu}^{2}\) Operator on Complete Riemannian Manifolds

\(\mathfrak{L}_{\nu}\) $${\mathfrak{L}}_{\nu }$$ operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of \(\mathfrak{L}_{\nu}^{2}\) $${\mathfrak{L}}_{\nu }^2$$ operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of \(\mathfrak{L}_{\nu}^{2}\) $${\mathfrak{L}}_{\nu }^2$$ operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the \(\mathcal{L}_{II}\) $${\mathcal{L}}_{II}$$ operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate.