Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds

Estimates for the Eigenvalues of the Drifting Laplacian on Some Complete Ricci Solitons

“…Those eigenvalue inequalities are related to the work from [37,39]. However, seen from the inequalities (1.7) and (1.8), our results are independent of the mean curvature.…”

“…Based on research from those examples, we can obtain the estimates for eigenvalues of the drifting Laplacian on some complete, rotationally invariant shrinking solitons. In order to prove Theorem 1.2, we need the following general formula established by Xia and Xu in [37].…”

First Eigenvalues of Geometric Operators Under the Yamabe Flow

“…As an extension of the Laplace-Beltrami operator, many classical results in Riemannian geometry asserted in terms of the LaplaceBeltrami operator have been extended to the analogous ones on the WittenLaplacian operator. For example, we can see these results ( [3], [4], [8], and [14]). Inspired by Cao [1] and Zhao [15], we study the first eigenvalue of the geometric operator −∆ φ + R 2 under the Yamabe flow.…”

Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons

“…A natural question is "Could we generalize those universal inequalities for the eigenvalue problem (1.1) on Riemannian manifolds to the case of the eigenvalue problem (1.2) on smooth metric measure spaces?" In fact, when m = 1, Xia and Xu [34] investigated the eigenvalues of the Dirichlet problem of the drifting Laplacian on compact manifolds and got some universal inequalities; when m = 2, Du et al [16] obtained some universal inequalities of Yang type for eigenvalues of the bi-drifting Laplacian problem either on a compact Riemannian manifold with boundary (possibly empty) immersed in a Euclidean space, a unit sphere or a projective space, or on bounded domains of complete manifolds supporting some special function. Recently, when m is an arbitrary integer no less than 2, Pereira et al [28] have given some universal inequalities on bounded domains in a Euclidean space or a unit sphere.…”

“…Recent years, a lot of universal inequalities for eigenvalues of Dirichlet and clamped plate problems on Riemannian manifolds have been obtained. We refer to [1,[3][4][5][9][10][11][19][20][21][22]27,[30][31][32][33][34][35] and the references therein. A smooth metric measure space (also known as the weighted measure space) is actually a Riemannian manifold equipped with some measure which is conformal to the usual Riemannian measure.…”