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In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with $Ric\geq (n-1)\kappa$, $\, \kappa\geq 0$, which extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian., Comment: Comments are welcome

In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse H\"older inequality for eigenfunctions of Dirichlet Laplacian., Comment: 16 pages. Comments are welcome

In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obtain the CR Obata theorem in a compact weighted Sasakian manifold with or without boundary.

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant.In two cases the Poincare-Einstein metrics are standard hyperbolic metric., Comment: 18 pages, all comments welcome

In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordstr\"om-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which can be realized as graphs over Reissner-Nordstr\"om-anti-deSitter manifold., Comment: 26 pages. All comments are welcome

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Bbb R$. For a positive function $\psi$ defined on $\bar M$, we generalized the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $\sigma_k(\kappa)=\psi(V,\nu(V))$. We also obtain some existence results for the starshaped compact hypersurface $\Sigma$ satisfying the above equation with various assumptions.

This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel, where $\Gamma_j$ denotes the $j^{^{\text{th}}}$ eigenvalue of the clamped plate problem.

In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$., Comment: the title is changed. to appear in Calculus of Variations and PDE's

In this paper, we establish some lower bounds for the sums of eigenvalues of the polyharmonic operator and higher order Stokes operator, which are sharper than the recent results in \cite{CSWZ13, I13}. At the same time, we obtain some certain bounds for the sums of positive and negative powers of eigenvalues of the polyharmonic operator., Comment: 14 pages. All comments are welcome

In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues., Comment: 19 pages,any comments are welcome

By the calculation of the gap of the consecutive eigenvalues of $\Bbb S^n$ with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal if this conjecture holds, obviously. In this paper, using new method, we solve this conjecture in the Euclidean space case intrinsically. We think our method is valid for the case of general Riemannian manifolds and give some examples directly., Comment: 15 pages

In this article we study surfaces in $\mathbb{S}^3(1) \times \mathbb{R}$ for which the $\mathbb{R}$-direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature vector., Comment: 16 pages

In this note, we obtain the sharp estimates for the first eigenvalue of Paneitz operator for $4$-dimensional compact submanifolds in Euclidean space. Since unit spheres and projective spaces can be canonically imbedded into Euclidean space, the corresponding estimates for the first eigenvalue are also obtained.

For $n\geq 7$, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact $n$-dimensional submanifolds in an $(n+p)$-dimensional space form.

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &\Delta {\mathbf u}+ \alpha{\rm grad}(\text{div}{\mathbf u})=-\sigma {\mathbf u}, \ \text{in $\Omega$}, &{\mathbf u}|_{\partial \Omega}={\mathbf 0}. \aligned . $$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the $(k+1)^{\text{th}}$ eigenvalue $\sigma_{k+1}$. We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature., Comment: 19 pages. This is a revised version. Since the third and forth authors also prove similar results to the first version. We decide to write this joint work

For a compact spin manifold $M$ isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the Dirac operators, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues., Comment: 12pages,submitted to "Math.Z.", Jun 2006

Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary., Comment: 12pages