first eigenvalue of a closed manifold with positive Ricci (original) (raw)

Estimate and monotonicity of the first eigenvalue under the Ricci flow

Mathematische Annalen, 2012

In this paper, we first derive a monotonicity formula for the first eigenvalue of − + a R (0 < a ≤ 1/2) 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.

Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature

Advances in Mathematics, 2000

We generalise for a general symmetric elliptic operator the different notions of dimension, diameter, and Ricci curvature, which coincide with the usual notions in the case of the Laplace Beltrami operators on Riemannian manifolds. If * 1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on * 1 one can obtain under an upper bound on the dimension, an upper bound on the diameter, and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R>0, then * 1 nRÂ(n&1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong Yang's estimate asserts that * 1 ? 2 d 2 , where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for * 1 which unifies and sharpens Lichnerowicz and Zhong Yang's estimates, together with other comparison results concerning the range of the associated eigenfunctions and their derivatives.

The first eigenvalue of the Laplacian on manifolds of nonnegative curvature

Proceedings of Symposia in Pure Mathematics, 1975

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Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature

For a complete Riemannian manifold (M, g) with nonpositive scalar curvature and a suitable domain Ω ⊂ M , let λ(Ω) be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on Ω. We prove several bounds for the rate of decrease of λ(Ω) as Ω increases, and a result comparing the rate of decrease of λ before and after a conformal diffeomorphism. Along the way, we obtain a reverse-Hölder inequality for the first eigenfunction, which generalizes results of Chiti to the manifold setting and may be of independent interest.

Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds

Communications in Analysis and Geometry, 2021

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

first eigenvalue of a closed manifold with positive Ricci (original) (raw)

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