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

Chen, Mu-Fa; Wang, Feng-Yu

doi:10.1007/bf02911438

Cited by 70 publications

(79 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…However, only the Lichnerowicz's estimate (1) was proved by J. F. Escobar until 1990. Except this, the others in (2)- (8) (and furthermore (9)- (12)) are all new in geometry [6] . (3) For more general non-compact manifolds, elliptic operators or Markov chains, we also have the corresponding dual variational formula [7], [8] .…”

Section: Theorem [General Formula] (Chen and Wangmentioning

confidence: 91%

See 1 more Smart Citation

Eigenvalues, Inequalities, and Ergodic Theory

Chen

2005

Probability and Its Applications

View full text Add to dashboard Cite

This paper surveys the main results obtained during the period 1992-1999 on three aspects mentioned at the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e., the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains ( §1). Here, a probabilistic method-coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger's method which comes from Riemannian geometry. This consists of §2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented ( §3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.

show abstract

Section: Theorem [General Formula] (Chen and Wangmentioning

confidence: 91%

“…All together, there are five sharp estimates ((1), (2), (4), (6) and (7)). The first two are sharp for the unit sphere in two-or higherdimension but it fails for the unit circle; the fourth, the sixth and the seventh estimates are all sharp for the unit circle.…”

mentioning

confidence: 99%

Eigenvalues, Inequalities, and Ergodic Theory

Chen

2005

Probability and Its Applications

View full text Add to dashboard Cite

show abstract

“…Recently Hang and Wang [2007] proved that equality (9) holds if and only if M is a circle or a segment. For related work see [Kröger 1992;Chen and Wang 1997;Bakry and Qian 2000]. These results were generalized to the p-Laplacian in [Valtorta 2012] and to the Laplacian on Alexandrov spaces in [Qian et al 2012].…”

Section: Introductionmentioning

confidence: 99%

“…Instead, we adopt the technique based on gradient comparison with a one dimensional model function, which was developed in [Kröger 1992] and improved in [Chen and Wang 1997;Bakry and Qian 2000]. Surprisingly, we find that the one dimensional model coincides with that for the Laplacian case.…”

Section: Introductionmentioning

confidence: 99%

An optimal anisotropic Poincaré inequality for convex domains

Wang¹,

Xia²

2012

Pacific J. Math.

View full text Add to dashboard Cite

In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of the anisotropic Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincaré inequality for convex domains, which generalizes the classical result of Payne and Weinberger. A lower bound of the first (nonzero) eigenvalue of the anisotropic Laplacian with the Dirichlet boundary condition is also proved.

show abstract

“…Besides Lichnerwicz's argument and Bakry-Emery's semigroup method developed for the positive curvature case, two general tools in the study of λ 1 are the maximal principle due to Li and Yau [12] and the coupling method due to Chen and the author [6], see e.g. [1], [11] and [13] for further developments of the first tool as well as [5], [7], [15] and [18] for update results based on the second.…”

Section: Introductionmentioning

confidence: 99%

Estimates of the first Neumann eigenvalue and the log‐Sobolev constant on non‐convex manifolds

Wang

2007

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on non-convex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obtained in the convex case. This method also works for more general functional inequalities. In particular, some explicit lower bounds are presented for the log-Sobolev constant on non-convex manifolds.

show abstract

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

Cited by 70 publications

References 5 publications

Eigenvalues, Inequalities, and Ergodic Theory

Eigenvalues, Inequalities, and Ergodic Theory

An optimal anisotropic Poincaré inequality for convex domains

Estimates of the first Neumann eigenvalue and the log‐Sobolev constant on non‐convex manifolds

Contact Info

Product

Resources

About