Eigenvalue Inequalities for the Clamped Plate Problem of ${\mathfrak{L}}_{\nu }^2$ Operator

${\mathfrak{L}}_{II}$ operator is introduced by Y.-L. Xin (Calculus of Variations and Partial Differential Equations. 2015, \textbf54(2):1995-2016), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend ${\mathfrak{L}}_{II}$ operator to more general elliptic differential operator ${\mathfrak{L}}_{\nu }$, and investigate the clamped plate problem of bi-${\mathfrak{L}}_{\nu }$ operator, which is denoted by ${\mathfrak{L}}_{\nu }^2$, on the complete Riemannian manifolds. A general formula of eigenvalues for the ${\mathfrak{L}}_{\nu }^2$ operator is established. Applying this formula, we estimate the eigenvalues with lower order on the Riemannian manifolds. As some further applications, we establish some eigenvalue inequalities for this operator on the translating solitons with respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit spheres and projective spaces. In particular, for the case of translating solitons, all of the eigenvalue inequalities are universal.