Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds

In this paper, we study eigenvalues of elliptic operators in divergence form on compact Riemannian manifolds with boundary (possibly empty) and obtain a general inequality for them. By using this inequality, we prove universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.     

Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds

In this paper, we investigate the Dirichlet eigenvalue problems of second order elliptic operators in divergence form on bounded domains of complete Riemannian manifolds. We discuss the cases of submanifolds immersed in a Euclidean space, Riemannian manifolds admitting spherical eigenmaps, and Riemannian manifolds which admit l functions ${f_\alpha : M \longrightarrow \mathbb{R}}$ such that ${\langle \nabla f_\alpha, \nabla f_\beta \rangle = \delta_{\alpha \beta}}$ and Δf _α = 0, where ? is the gradient operator. Some inequalities for lower order eigenvalues of these problems are established. As applications of these results, we obtain some universal inequalities for lower order eigenvalues of the Dirichlet Laplacian problem. In particular, the universal inequality for eigenvalues of the Laplacian on a unit sphere is optimal.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.     

Sobolev type inequalities on Riemannian manifolds

This paper studies Sobolev type inequalities on Riemannian manifolds. We show that on a complete non-compact Riemannian manifold the constant in the Gagliardo-Nirenberg inequality cannot be smaller than the optimal one on the Euclidean space of the same dimension. We also show that a complete non-compact manifold with asymptotically non-negative Ricci curvature admitting some Gagliardo-Nirenberg inequality is not very far from the Euclidean space.     

Characterizations of umbilic hypersurfaces in warped product manifolds

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.     

带权Paneitz算子和带权圆盘振动问题的特征值不等式

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz算子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界估计.     

Some inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues in bounded domains of Riemannian manifolds. We also obtain the Weyl-type asymptotic formulas for the eigenvalues of the buckling and clamped plate problems in bounded domains of Riemannian manifolds. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds

 In this paper we study warped product CR-submanifolds in Kaehler manifolds and introduce the notion of CR-warped products. We prove several fundamental properties of CR-warped products in Kaehler manifolds and establish a general inequality for an arbitrary CR-warped product in an arbitrary Kaehler manifold. We then investigate CR-warped products in a general Kaehler manifold which satisfy the equality case of the inequality. Finally we classify CR-warped products in complex Euclidean space which satisfy the equality. (Received 24 August 2000; in revised form 19 February 2001)     

Eigenvalue inequalities for the clamped plate problem of $${\mathfrak{L}}_\nu ^2$$ operator

\({{\mathfrak{L}}_{II}}\) operator is introduced by Xin (2015), 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 a 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 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.

     

On Riemannian manifolds foliated by (<Emphasis Type="Italic">n</Emphasis> − 1)-umbilical hypersurfaces

In this paper we define closed partially conformal vector fields and use them to give a characterization of Riemannian manifolds which admit this kind of fields as some special warped products foliated by (n − 1)-umbilical hypersurfaces. Examples are described in space forms. In particular, closed partially conformal vector fields in Euclidean spaces are associated to the most simple foliations given by hyperspheres, hyperplanes or coaxial cylinders. Finally, for manifolds admitting such vector fields, we impose conditions for a hypersurface to be (n − 1)-umbilical, or, in particular, a leaf of the corresponding foliation.     

Geometric classification of warped products isometrically immersed into Sasakian space forms

The main objective of this paper is to study the warped product pointwise semi‐slant submanifolds which are isometrically immersed into Sasakian manifolds. First, we prove some characterizations results in terms of the shape operator, under which influence a pointwise semi‐slant submanifold of a Sasakian manifold can be reduced to a warped product submanifold. Then, we determine a geometric inequality for the second fundamental form regarding to intrinsic invariant and extrinsic invariant using the Gauss equation instead of the Codazzi equation. Evenmore, we give some applications of this inequality into Sasakian space forms, and we will investigate the status of equalities in the inequality. As a particular case, we provide numerous applications of the Green lemma, the Laplacian of warped functions and some partial differential equations. Some triviality results for connected, compact warped product pointwise semi‐slant submanifolds of Sasakian space form by means of Hamiltonian and the kinetic energy of warped function involving boundary conditions are established.     

Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

     

UNIVERSAL INEQUALITIES FOR A HORIZONTAL LAPLACIAN VERSION OF THE CLAMPED PLATE PROBLEM ON CARNOT GROUP

In this paper, we investigate a horizontal Laplacian version of the clamped plate problem on Carnot groups and obtain some universal inequalities. Furthermore, for the lower order eigenvalues of this eigenvalue problem on carnot groups, we also give some universal inequalities.     

Doubly warped product CR -submanifolds in locally conformal K?hler manifolds

In this paper we study doubly warped product CR submanifolds in locally conformal K?hler manifolds, and we found a B.Y. Chen’s type inequality for the second fundamental form of these submanifolds.     

Constant mean curvature surfaces in warped product manifolds

We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.     

Eigenvalue inequalities for the buckling problem of the drifting Laplacian

In this paper, we study the buckling problem of the drifting Laplacian on bounded domains in a complete Riemannian manifold with nonnegative ∞-dimensional Bakry–Émery Ricci curvature. According to the property of the manifold, we obtain a family of trial functions. By making use of these trial functions, we derive a universal inequality of eigenvalues, which is independent of the domains.     

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.     

Estimates for lower order eigenvalues of a clamped plate problem

For a bounded domain Ω in a complete Riemannian manifold M ⁿ , we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.     

Trudinger‐Moser inequalities on the entire Heisenberg group

Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis, 2011), we obtain a class of Trudinger‐Moser inequalities on the entire Heisenberg group, which indicate what the best constants are. All the existing proofs of similar inequalities on unbounded domain of the Euclidean space or the Heisenberg group are based on rearrangement argument. In this note, we propose a new approach to solve this problem. Specifically we get the global Trudinger‐Moser inequality by gluing local estimates with the help of cut‐off functions. Our method still works for similar problems when the Heisenberg group is replaced by the Euclidean space or complete noncompact Riemannian manifolds.