  $\mathfrak{L}_{\nu}$ 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 $\mathfrak{L}^{2}_{\nu}$ operator on a
bounded domain of the complete Riemannian manifolds. A general formula of
eigenvalues of $\mathfrak{L}^{2}_{\nu}$ operator is established. Applying this
general formula, we obtain some estimates for the eigenvalues with higer 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 $\mathcal{L}_{II}$ 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 inequality.
