Small eigenvalues and thick-thin decomposition in negative curvature

Hamenstädt, Ursula

doi:10.5802/aif.3345

Cited by 4 publications

(10 citation statements)

References 16 publications

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“…By the main result of , the smallest positive eigenvalue of the thick part of a mapping torus with Neumann boundary conditions can be estimated in the same way. Theorem follows from this fact and as explained in Section . The proof of Theorem for typical mapping tori is contained in Section .…”

Section: Introductionmentioning

confidence: 67%

“…If

M

is uniformly quasi‐isometric to a finite graph

G

, then the smallest positive eigenvalue of

M

is uniformly equivalent to the smallest positive eigenvalue of

G

. This statement is also valid without modification for compact manifolds

M

with boundary and Neumann boundary conditions (see for a more precise statement).…”

Section: Arrays Of Circlesmentioning

confidence: 82%

“…Let us assume that for some fixed number

c_{1} \in false(0, 1 false)

, the translation length for the action of

ϕ

on the curve graph of the surface of genus

g

is at least

c_{1} ℓ

. Then the length of the base circle of the array of circles constructed from

M

in the proof of Proposition is at least

c_{2} vol false(M_{thick} false) = c_{3} vol false(M false)

where

c_{2}, c_{3} > 0

only depend on

c

and

g

(compare the discussion in for a comparison between

vol (M_{thick})

and

vol (M)

).…”

Section: The Thick Part Of a Mapping Torusmentioning

confidence: 99%

“…The existence of expander families yield that the dependence of

b_{4} false(g, ε false)

g

is necessary. We refer to for a more complete discussion.…”

Section: Introductionmentioning

confidence: 99%

“…We estimate effectively the smallest eigenvalue

λ_{1} false(M_{thick} false)

M_{thick}

with Neumann boundary conditions as a function of the volume. We then use a result of : There exists a universal constant

b > 0

such that

b^{- 1} λ_{1} false(M_{thick} false) ⩽ λ_{1} false(M false) ⩽ b \log vol false(M_{thin} false) λ_{1} false(M_{thick} false)

for every closed hyperbolic 3‐manifold

M

. The factor

\log vol (M)

in the statement of the first part of Theorem arises from the ratio

λ_{1} false(M false) / λ_{1} false(M_{thick} false)

.…”

Section: Introductionmentioning

confidence: 99%

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The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

Baik

Gekhtman

Hamenstädt

2019

Proc. London Math. Soc.

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We study the smallest positive eigenvalue λ1false(Mfalse) of the Laplace–Beltrami operator on a closed hyperbolic 3‐manifold M which fibers over the circle, with fiber a closed surface of genus g⩾2. We show the existence of a constant C>0 only depending on g so that λ1false(Mfalse)∈false[C−1/ vol false(Mfalse)2,Clog vol (M)/ vol false(Mfalse)22g−2/(22g−2−1)false] and that this estimate is essentially sharp. We show that if M is typical or random, then we have λ1false(Mfalse)∈false[C−1/ vol false(Mfalse)2,C/ vol false(Mfalse)2false]. This rests on a result of independent interest about reccurence properties of axes of random pseudo‐Anosov elements.

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Section: Introductionmentioning

confidence: 67%

“…If

M

is uniformly quasi‐isometric to a finite graph

G

, then the smallest positive eigenvalue of

M

is uniformly equivalent to the smallest positive eigenvalue of

G

. This statement is also valid without modification for compact manifolds

M

with boundary and Neumann boundary conditions (see for a more precise statement).…”

Section: Arrays Of Circlesmentioning

confidence: 82%

“…Let us assume that for some fixed number

c_{1} \in false(0, 1 false)

, the translation length for the action of

ϕ

on the curve graph of the surface of genus

g

is at least

c_{1} ℓ

. Then the length of the base circle of the array of circles constructed from

M

in the proof of Proposition is at least

c_{2} vol false(M_{thick} false) = c_{3} vol false(M false)

where

c_{2}, c_{3} > 0

only depend on

c

and

g

(compare the discussion in for a comparison between

vol (M_{thick})

and

vol (M)

).…”

Section: The Thick Part Of a Mapping Torusmentioning

confidence: 99%

“…The existence of expander families yield that the dependence of

b_{4} false(g, ε false)

g

is necessary. We refer to for a more complete discussion.…”

Section: Introductionmentioning

confidence: 99%

“…We estimate effectively the smallest eigenvalue

λ_{1} false(M_{thick} false)

M_{thick}

with Neumann boundary conditions as a function of the volume. We then use a result of : There exists a universal constant

b > 0

such that

b^{- 1} λ_{1} false(M_{thick} false) ⩽ λ_{1} false(M false) ⩽ b \log vol false(M_{thin} false) λ_{1} false(M_{thick} false)

for every closed hyperbolic 3‐manifold

M

. The factor

\log vol (M)

in the statement of the first part of Theorem arises from the ratio

λ_{1} false(M false) / λ_{1} false(M_{thick} false)

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

Baik

Gekhtman

Hamenstädt

2019

Proc. London Math. Soc.

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On the boundary injectivity radius of Buser–Colbois–Dodziuk-Margulis tubes

Cerbo

Luca²

2021

Ann Glob Anal Geom

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Small eigenvalues of random 3-manifolds

Hamenstädt

Viaggi

2022

Trans. Amer. Math. Soc.

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We show that for every g ≥ 2 g\geq 2 there exists a number c = c ( g ) > 0 c=c(g)>0 such that the smallest positive eigenvalue of a random closed 3-manifold M M of Heegaard genus g g is at most c ( g ) / v o l ( M ) 2 c(g)/{\mathrm {vol}}(M)^2 .

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Small eigenvalues and thick-thin decomposition in negative curvature

Abstract: Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org

Cited by 4 publications

References 16 publications

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

On the boundary injectivity radius of Buser–Colbois–Dodziuk-Margulis tubes

Small eigenvalues of random 3-manifolds

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