Lingzhong Zeng

${\mathfrak{L}}_{II}$ 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.

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigenvalue problems, which is sharp in the sense of the order of the eigenvalues. As some interesting applications, we study the eigenvalue of drifting Laplacian on Ricci solitons, self-shrinkers and product Riemannian manifolds. We give the explicit upper bounds of the gap of the consecutive eigenvalues of the drifting Laplacian. Since eigenvalues is invariant in the sense of isometry, by the classifications of Ricci solitons and self-shrinkers, we give the explicit upper bounds for the consecutive eigenvalues of the drifting Laplacian on a large class metric measure spaces. In addition, we also consider the case of product Riemannian manifolds with certain curvature conditions and some upper bounds are obtained. Basing on the case of Laplace operator, we also present a conjecture as follows: all of the eigenvalues of the Dirichlet problem of drifting Laplacian on metric measure spaces satisfy: $${\lambda }_{k+1}-{\lambda }_k\le ({\lambda }_2-{\lambda }_1)k^{\frac{1}{n}}.$$We note the conjecture is true in some special cases.