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

TL;DR

This paper derives eigenvalue formulas and estimates for the $rak{L}_ u^2$ operator's clamped plate problem on complete Riemannian manifolds, with applications to various geometric submanifolds and a new universal inequality.

Contribution

It provides a general eigenvalue formula and estimates for the $rak{L}_ u^2$ operator, extending spectral analysis to complex geometric settings and establishing a universal inequality.

Findings

Established a general eigenvalue formula for $rak{L}_ u^2$ operator.

Derived eigenvalue estimates for higher order on Riemannian manifolds.

Proposed a universal inequality for the $rak{L}_{II}$ operator on translating solitons.

Abstract

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