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

Eldredge, Nathaniel; Gordina, Maria; Saloff‐Coste, Laurent

doi:10.1007/s00039-018-0457-8

Cited by 12 publications

(21 citation statements)

References 51 publications

(85 reference statements)

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“…The expression for

λ_{1} false(prefixSU (2), g_{false(a, b, c false)} false)

in Theorem implies the following result, which refines [, Corollary 8.6] by giving optimal values to the bounds. The case

SO (3)

is also considered.…”

Section: Estimates On Homogeneous 3‐spheressupporting

confidence: 54%

“…This method requires showing the existence of a uniform upper bound of the volume doubling constant among the space of left‐invariant metrics. Moreover, they proved (among many other things) that this condition holds for

G = SU (2)

, and therefore the above conjecture holds in this case (see [, Theorem 8.5]).…”

Section: Introductionmentioning

confidence: 85%

“…Moreover, this lower bound is universal in the sense that it does not depend on

M

. Concerning upper bounds, Eldredge, Gordina and Saloff‐Coste recently made the following conjecture (see [, (1.2)]). Conjecture For each compact Lie group

G

, there is a positive real number

C_{G}

such that, for any left‐invariant metric

g

G

, one has

λ_{1} false(G, g false) ⩽ \frac{C_{G}}{diam {(G, g)}^{2}} .

…”

Section: Introductionmentioning

confidence: 91%

“…However, we will bound it by expressions in terms of the parameters

a

b

and

c

. The proof is based on the proof of [, Proposition 7.1], which shows that

diam (SU false(2 false), g_{(a, b, c)})

is comparable with

b^{- 1}

for all

a ⩾ b ⩾ c > 0

. That is, there are positive real numbers

D_{0}

and

D_{\infty}

such that

D_{0} b^{- 1} ⩽ diam false(prefixSU (2), g_{false(a, b, c false)} false) ⩽ D_{\infty} b^{- 1} .

We will refine the above bounds for

diam (G, g_{(a, b, c)})

(see Proposition ), obtaining as a consequence explicit and optimal values for

D_{0}

and

D_{\infty}

(see Corollary ).…”

Section: Estimates On Homogeneous 3‐spheresmentioning

confidence: 97%

See 3 more Smart Citations

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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“…The expression for

λ_{1} false(prefixSU (2), g_{false(a, b, c false)} false)

in Theorem implies the following result, which refines [, Corollary 8.6] by giving optimal values to the bounds. The case

SO (3)

is also considered.…”

Section: Estimates On Homogeneous 3‐spheressupporting

confidence: 54%

G = SU (2)

, and therefore the above conjecture holds in this case (see [, Theorem 8.5]).…”

Section: Introductionmentioning

confidence: 85%

“…Moreover, this lower bound is universal in the sense that it does not depend on

M

. Concerning upper bounds, Eldredge, Gordina and Saloff‐Coste recently made the following conjecture (see [, (1.2)]). Conjecture For each compact Lie group

G

, there is a positive real number

C_{G}

such that, for any left‐invariant metric

g

G

, one has

λ_{1} false(G, g false) ⩽ \frac{C_{G}}{diam {(G, g)}^{2}} .

…”

Section: Introductionmentioning

confidence: 91%

“…However, we will bound it by expressions in terms of the parameters

a

b

and

c

. The proof is based on the proof of [, Proposition 7.1], which shows that

diam (SU false(2 false), g_{(a, b, c)})

is comparable with

b^{- 1}

for all

a ⩾ b ⩾ c > 0

. That is, there are positive real numbers

D_{0}

and

D_{\infty}

such that

D_{0} b^{- 1} ⩽ diam false(prefixSU (2), g_{false(a, b, c false)} false) ⩽ D_{\infty} b^{- 1} .

We will refine the above bounds for

diam (G, g_{(a, b, c)})

(see Proposition ), obtaining as a consequence explicit and optimal values for

D_{0}

and

D_{\infty}

(see Corollary ).…”

Section: Estimates On Homogeneous 3‐spheresmentioning

confidence: 97%

See 2 more Smart Citations

The smallest Laplace eigenvalue of homogeneous 3-spheres

Lauret

2018

Bull. London Math. Soc.

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show abstract

“…Following Nielsen, we will consider I IJ s that are diagonal in the generalized Pauli basis and for which the penalty factor of a given generalized Pauli is solely a function of the k-locality, i.e. solely a function of the weight k (or size 13 ) of the operator, defined as the number of capital indices in Eq. 11.…”

Section: Review Of Complexity Geometrymentioning

confidence: 99%

Effective Geometry, Complexity, and Universality

Brown¹,

Freedman²,

Lin³

et al. 2021

Preprint

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Post-Wilsonian physics views theories not as isolated points but elements of bigger universality classes, with effective theories emerging in the infrared. This paper makes initial attempts to apply this viewpoint to homogeneous geometries on group manifolds, and complexity geometry in particular. We observe that many homogeneous metrics on low-dimensional Lie groups have markedly different short-distance properties, but nearly identical distance functions at longer distances. Using Nielsen's framework of complexity geometry, we argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered in complexity theory. We conjecture that at larger complexities, a new effective metric emerges that describes a broad class of complexity geometries, insensitive to various choices of 'ultraviolet' penalty factors. Finally we lay out a broader mathematical program of classifying the effective geometries of right-invariant group manifolds.

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Diameter and Laplace Eigenvalue Estimates for Compact Homogeneous Riemannian Manifolds

LAURET

2022

Transformation Groups

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

Cited by 12 publications

References 51 publications

The smallest Laplace eigenvalue of homogeneous 3-spheres

The smallest Laplace eigenvalue of homogeneous 3-spheres

Effective Geometry, Complexity, and Universality

Diameter and Laplace Eigenvalue Estimates for Compact Homogeneous Riemannian Manifolds

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