Upper bounds for the spectral function on homogeneous spaces via volume growth

Judge, Chris; Lyons, Russell

doi:10.4171/rmi/1103

Cited by 8 publications

(7 citation statements)

References 21 publications

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“…As mentioned earlier, the lower bound was proved in [34]. (An improved lower bound was recently obtained in [28].) To obtain an upper bound, we note that…”

Section: The Poincaré Inequality On Compact Lie Groupsmentioning

confidence: 92%

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Left-invariant geometries on SU(2) are uniformly doubling

Eldredge¹,

Gordina²,

Saloff‐Coste³

2018

Geom. Funct. Anal.

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A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on SU(2). We show that left-invariant geometries on SU(2) are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.1991 Mathematics Subject Classification. Primary 53C21; Secondary 35K08, 53C17, 58J35, 58J60, 22C05, 22E30.

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“…As mentioned earlier, the lower bound was proved in [34]. (An improved lower bound was recently obtained in [28].) To obtain an upper bound, we note that…”

Section: The Poincaré Inequality On Compact Lie Groupsmentioning

confidence: 92%

“…When (M, g) is a compact homogeneous space, C. Judge and R. Lyons in [28] have recently obtained the following uniform upper bound.…”

Section: The Poincaré Inequality On Compact Lie Groupsmentioning

confidence: 99%

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Left-invariant geometries on SU(2) are uniformly doubling

Eldredge¹,

Gordina²,

Saloff‐Coste³

2018

Geom. Funct. Anal.

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“…Recently, Judge and Lyons [JL19] improved it. Recall that a Riemannian manifold is called homogeneous if its isometry group acts transitively on it.…”

Section: Introductionmentioning

confidence: 99%

Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

Lauret

2020

Preprint

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Let G be a compact connected Lie group of dimension m. Once a bi-invariant metric on G is fixed, left-invariant metrics on G are in correspondence with m × m positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on G in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets S of the space of left-invariant metrics M on G such that there exists a positive real number C depending on G and S such that λ 1 (G, g) diam(G, g) 2 ≤ C for all g ∈ S. The existence of the constant C for S = M is the original conjecture.

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“…During the review process, the author found the recent article by Judge and Lyons, which improves the lower bound in .…”

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confidence: 99%

The smallest Laplace eigenvalue of homogeneous 3-spheres

Lauret

2018

Bull. London Math. Soc.

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We establish an explicit expression for the smallest non‐zero eigenvalue of the Laplace–Beltrami operator on every homogeneous metric on the 3‐sphere, or equivalently, on SU(2) endowed with left‐invariant metric. For the subfamily of three‐dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non‐zero eigenvalue in terms of the diameter for homogeneous 3‐spheres. Another application shows that the spectrum of the Laplace–Beltrami operator distinguishes up to isometry any left‐invariant metric on SU(2). It is also proved the non‐existence of constant scalar curvature metrics conformal and arbitrarily close to any non‐round homogeneous metric on the 3‐sphere. All of the above results are extended to left‐invariant metrics on SO(3), that is, homogeneous metrics on the three‐dimensional real projective space.

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Upper bounds for the spectral function on homogeneous spaces via volume growth

Cited by 8 publications

References 21 publications

Left-invariant geometries on SU(2) are uniformly doubling

Left-invariant geometries on SU(2) are uniformly doubling

Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

The smallest Laplace eigenvalue of homogeneous 3-spheres

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