Eigenvalues of Xin-Laplacian on Complete Riemannian manifolds

TL;DR

This paper investigates eigenvalues of the Xin-Laplacian on complete Riemannian manifolds, establishing inequalities, exploring special cases like translating solitons, and providing sharp bounds for eigenvalues on minimal submanifolds.

Contribution

It introduces new eigenvalue inequalities for the Xin-Laplacian, extends Reilly's result, and offers sharp estimates for eigenvalues on specific minimal submanifolds, broadening understanding of spectral geometry.

Findings

Established general formulas for eigenvalues of Xin-Laplacian.

Proved universal eigenvalue inequalities for translating solitons.

Derived sharp bounds for the second eigenvalue on minimal isoparametric hypersurfaces.

Abstract

In this paper, we firstly consider Dirichlet eigenvalue problem which is related to Xin-Laplacian on the bounded domain of complete Riemannian manifolds. By establishing the general formulas, combining with some results of Chen and Cheng type, we prove some eigenvalue inequalities. As some applications, we consider the eigenvalues on some Riemannian manifolds admitting with special functions, the translating solitons, minimal submanifolds on the Euclidean spaces, submanifolds on the unit spheres, projective spaces and so on. In particular, for the case of translating solitons, some eigenvalue inequalities are universal. Moreover, we investigate the closed eigenvalue problem for the Xin-Laplacian and generalize the Reilly's result on the first eigenvalue of the Laplace-Beltrami operator. As some remarkable applications, we obtain a very sharp estimate for the upper bound of the second nonzero eigenvalue(without counting multiplicities of eigenvalues) of the Laplace-Beltrami operator on the minimal isoparametric hypersurfaces and focal submanifolds in the unit sphere, which leads to a conjecture and is the most fascinating part of this paper.