On the first eigenvalue of the clamped plate

Summary We find a lower bound for the ratio between the first eigenvalue of any homogeneous thin plate G, which is clamped on its boundary, and the first eigenvalue of the spherical clamped plate having the same measure as G. In two dimensions, our bound is about 0.98.     

The first eigenvalue of a closed manifold with positive Ricci curvature

We give a new estimate on the lower bound for the first positive eigenvalue of the Laplacian on a closed manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the largest interior radius of the nodal domains of eigenfunctions of the eigenvalue.

     

Estimate and monotonicity of the first eigenvalue under the Ricci flow

In this paper, we first derive a monotonicity formula for the first eigenvalue of on a closed surface with nonnegative scalar curvature under the (unnormalized) Ricci flow. We then derive a general evolution formula for the first eigenvalue under the normalized Ricci flow. As an application, we obtain various monotonicity formulae and estimates for the first eigenvalue on closed surfaces.     

Time-dependent Sobolev inequality along the Ricci flow

In this article, we get a time-dependent Sobolev inequality along the Ricci flow in a more general situation than those in Zhang (A uniform Sobolev inequality under Ricci flow. Int Math Res Not IMRN 2007, no 17, Art ID rnm056, 17 pp), Ye (The logarithmic Sobolev inequality along the Ricci flow. arXiv:0707.2424v2) and Hsu (Uniform Sobolev inequalities for manifolds evolving by Ricci flow. arXiv:0708.0893v1) which also generalizes the results of them. As an application of the time-dependent Sobolev inequality, we get a growth of the ratio of non-collapsing along immortal solutions of Ricci flow.     

The Ricci flow as a geodesic on the manifold of Riemannian metrics

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.     

First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow

In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator (p ?? 2) along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma??s results [Ann. Glob. Anal. Geom., 29, 287?C292 (2006)].     

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 probabilistic approach to the first Dirichlet eigenvalue on non-compact Riemannian manifold

By using diffusion process with absorbing boundary, some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+∇h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean's estimate for ∇h=0. Moreover, the first Dirichlet eigenvalue for elliptic operators onR ^d and the first mixed eigenvalue are also studied. Some examples show that our estimates can be sharp even for ∇h≠0. Research supported in part by NFSC and the State Education Commission of China     

On the first eigenvalue of Finsler manifolds with nonnegative weighted Ricci curvature

We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(resp.a segment).Moreover,a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated.These generalize the corresponding results in recent literature.     

The second Sobolev best constant along the Ricci flow

In this work we present some properties satisfied by the second L ²-Riemannian Sobolev best constant along the Ricci flow on compact manifolds of dimensions n ≥ 4. We prove that, along the Ricci flow g(t), the second best constant B ₀(2, g(t)) depends continuously on t and blows-up in finite time. In certain cases, the speed of the explosion is, at least, the same one of the curvature operator. We also show that, on manifolds with positive curvature operator or pointwise 1/4-pinched curvature, one of the situations holds: B ₀(2, g(t)) converges to an explicit constant or extremal functions there exists for t large.      

The first Dirichlet eigenvalue of a compact manifold and the Yang conjecture

We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution of Yamabe constant under Ricci flow

In this note under a crucial technical assumption, we derive a formula for the derivative of Yamabe constant , where g(t) is a solution of Ricci flow on closed manifold. We also give a simple application. Mathematics Subject Classifications (2000): 53C21 and 53C44     

Evolution and monotonicity of eigenvalues under the Ricci flow

Let(M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator-△φ+ c R under the Ricci flow and the normalized Ricci flow, where △φis the Witten-Laplacian operator, φ∈ C∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when c 14.     

Immortal solution of the Ricci flow

For any complete noncompact Kahler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.     

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature

Let L=Δ−∇φ⋅∇ be a symmetric diffusion operator with an invariant measure on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry-Émery Ricci curvature satisfying Ricm,n(L)?−(n−1), and therefore generalize a Cheng's result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.     

First eigenvalue of the <Emphasis Type="Italic">p</Emphasis>-Laplace operator along the Ricci flow

In this article, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the p-Laplace operator along the Ricci flow on closed manifolds. We show that the first p-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first p-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a p-eigenvalue comparison-type theorem when its Euler characteristic is negative.     

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.

     

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.