A sharp lower bound for the first eigenvalue on Finsler manifolds

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.     

The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold

This paper studies, using the Bochner technique, a sharp lower bound of the first eigenvalue of a subelliptic Laplace operator on a strongly pseudoconvex CR manifold in terms of its pseudo-Hermitian geometry. For dimensions greater than or equal to , the lower bound under a condition on the Ricci curvature and the torsion was obtained by Greenleaf. We give a proof for all dimensions greater than or equal to . For dimension , the sharp lower bound is proved under a condition which also involves a distinguished covariant derivative of the torsion.

     

The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold

In this paper, we study a sharp lower bound of the first eigenvalue of the sublaplacian on a 3-dimensional pseudohermitian manifold with the CR Paneitz operator positive. In general cases, S.-Y. Li and H.-S. Luk ({Proc. Am. Math. Soc.} 132(3), 789–798) (2004) proved the lower bound under a condition on a covariant derivative of the torsion as well as the Ricci curvature and the torsion. We show that if the CR Paneitz operator is positive, then the sharp lower bound is obtained under one simpler condition on only the Ricci curvature and the torsion itself; which is similar to the condition given in high-dimensional cases in ({Commun. Partial Differential Equations}, 10(2/3), 191–217) (1985). We also show examples where our theorem applies, but Theorem 1.2 in ({Proc. Am. Math. Soc.} 132(3), 789–798) (2004) does not. Mathematics Subject Classifications (2000). Primary 32V05, 32V20, Secondary 53C56.     

A lower bound for the first eigenvalue of a vibrating nonhomogeneous plate

Summary In this note we obtain a lower bound for the first eigenvalue of a nonhomogeneous plate problem. As a consequence we obtain an inequality of Barta-type.

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Résumé En cette note nous obtenons une borne inférieure pour la première valeur propre d'une plaque inhomogène. En conséquence nous obtenons une inégalité de type Barta.

     

A LOWER BOUND FOR FIRST EIGENVALUE WITH MIXED BOUNDARY CONDITIOIN

Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature RicM≥n- 1. The paper obtains an inequality for the first eigenvalue η1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz,Reilly, Escobar and Xia. It is also proved that η1≥ n for certain n-dimensional compact Riemannian manifolds with boundary,which is an extension of the work of Cheng,Li and Yau.     

A lower bound for the first eigenvalue of an elliptic operator

Oscillatory behavior of the solutions of the nth-order delay differential equation L_nx(t) + q(t)f(x[g(t)]) = 0 is discussed. The results obtained are extensions of some of the results by Kim (Proc. Amer. Math. Soc.62 (1977), 77–82) for y⁽ⁿ⁾ + py = 0.     

A spin-conformal lower bound of the first positive Dirac eigenvalue

Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. We prove the existence of a lower bound on the first positive eigenvalue of D depending only on the spin structure and the conformal type.     

General formula for lower bound of the first eigenvalue on Riemannian manifolds

A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’ s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).     

On a lower bound for the perron eigenvalue

A lower boundn ^–1 _i,k a_ik for the Perron eigenvalue of a symmetric non-negative irreducible matrixA=(a _ik) is studied and compared with certain other lower bounds.     

A sharp lower bound for the log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function ${\varphi}$ with an isolated singularity at 0 in an open subset of ${\mathbb{C}^n}$ . This threshold is defined as the supremum of constants c > 0 such that ${e^{-2c\varphi}}$ is integrable on a neighborhood of 0. We relate ${c(\varphi)}$ to the intermediate multiplicity numbers ${e_j(\varphi)}$ , defined as the Lelong numbers of ${(dd^c\varphi)^j}$ at 0 (so that in particular ${e_0(\varphi)=1}$ ). Our main result is that ${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$ . This inequality is shown to be sharp; it simultaneously improves the classical result ${c(\varphi)\geqslant 1/e_1(\varphi)}$ due to Skoda, as well as the lower estimate ${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.     

A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Pólya and Szegö's upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains which depends on the support function of the domain.

As a by-product, we also obtain a sharp upper bound for the spectral gap of convex domains.

     

A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

Let M be a closed hypersurface in a simply connected rank-1 symmetric space . In this paper, we give an upper bound for the first eigenvalue of the Laplacian of M in terms of the Ricci curvature of and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of M.     

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the \(p\) -Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator \(\Delta _p\) when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.     

A lower bound for the least eigenvalue of Δ+V

In this note we give a sufficient condition for Δ+V to be positive on a closed Riemannian manifold. We also give an application to a Bochner type vanishing theorem. This research was partially done under the E.E.C. Contract # SC 1-0105-C “GADGET” at the C.N.R.S. U.R.A. 188     

A sharp bound for the area of minimal surfaces in the unit ball

Let Σ be a k-dimensional minimal surface in the unit ball B ⁿ which meets the boundary ? B ⁿ orthogonally. We show that the area of Σ is bounded from below by the volume of the unit ball in ${\mathbb{R}^k}$ .     

An upper bound for the first eigenvalue of a spherical cap

By working with suitable test functions, we obtain an upper bound for the principal eigenvalue of a geodesic ball on a sphere of arbitrary dimension. This bound is sharp in the limiting case when the radius of the ball approaches the diameter of the sphere.This research was supported by ARO Grant 28905-MA.