Spectral bounds on closed hyperbolic 3-manifolds

White, Nina

doi:10.1112/jlms/jds082

Cited by 11 publications

(20 citation statements)

References 10 publications

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“…. , ρ k : M → R whose supports pairwise intersect on zero-volume sets, we have λ k max{R(ρ i ) | 0 i k} (compare [31]) and therefore λ 1 (M ) max{R(α • f ), R(β • f )}. As a result, λ 1 (M ) d/u 2 where d > 0 is a constant only depending on g, ε.…”

Section: The Smallest Eigenvalue Of Mapping Torimentioning

confidence: 94%

“…To this end, note that the Rayleigh quotients of

α, β

are

π^{2} / u^{2}

, and since

f

has uniformly bounded derivative, the Rayleigh quotients of

α, β

are uniformly equivalent to the Rayleigh quotients of

α \circ f, β \circ f

. By the Minmax theorem for the spectrum of the Laplacian, we know that for any set of functions

ρ_{0}, \dots, ρ_{k} : M \to R

whose supports pairwise intersect on zero‐volume sets, we have

λ_{k} ⩽ \max false{scriptR (ρ_{i}) ∣ 0 ⩽ i ⩽ k false}

(compare ) and therefore

λ_{1} false(M false) ⩽ \max false{scriptR (α \circ f), scriptR (β \circ f) false}

. As a result,

λ_{1} false(M false) ⩽ d / u^{2}

where

d > 0

is a constant only depending on

g, ε

.…”

Section: The Smallest Eigenvalue Of Mapping Torimentioning

confidence: 99%

“…We suspect that Theorem generalizes to hyperbolic mapping tori of nonexceptional surfaces of finite type with punctures, but we did not check the details. By the work of White , the injectivity radius of the examples in the second part of the above theorem tends to zero with

i

.…”

Section: Introductionmentioning

confidence: 95%

“…For manifolds

M

with a given lower bound of the injectivity radius, there is more precise information. Namely, White proved that there exists a number

b_{4} = b_{4} false(g, ε false) > 0

such that

λ_{1} false(M false) ⩽ \frac{b_{4} false(g, ε false)}{vol {(M)}^{2}}

for all closed hyperbolic 3‐manifolds of Heegaard genus at most

g

and injectivity radius at least

ε

. The existence of expander families yield that the dependence of

b_{4} false(g, ε false)

g

is necessary.…”

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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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.

show abstract

Section: The Smallest Eigenvalue Of Mapping Torimentioning

confidence: 94%

“…To this end, note that the Rayleigh quotients of

α, β

are

π^{2} / u^{2}

, and since

f

has uniformly bounded derivative, the Rayleigh quotients of

α, β

are uniformly equivalent to the Rayleigh quotients of

α \circ f, β \circ f

. By the Minmax theorem for the spectrum of the Laplacian, we know that for any set of functions

ρ_{0}, \dots, ρ_{k} : M \to R

whose supports pairwise intersect on zero‐volume sets, we have

λ_{k} ⩽ \max false{scriptR (ρ_{i}) ∣ 0 ⩽ i ⩽ k false}

(compare ) and therefore

λ_{1} false(M false) ⩽ \max false{scriptR (α \circ f), scriptR (β \circ f) false}

. As a result,

λ_{1} false(M false) ⩽ d / u^{2}

where

d > 0

is a constant only depending on

g, ε

.…”

Section: The Smallest Eigenvalue Of Mapping Torimentioning

confidence: 99%

i

.…”

Section: Introductionmentioning

confidence: 95%

“…For manifolds

M

with a given lower bound of the injectivity radius, there is more precise information. Namely, White proved that there exists a number

b_{4} = b_{4} false(g, ε false) > 0

such that

λ_{1} false(M false) ⩽ \frac{b_{4} false(g, ε false)}{vol {(M)}^{2}}

for all closed hyperbolic 3‐manifolds of Heegaard genus at most

g

and injectivity radius at least

ε

. The existence of expander families yield that the dependence of

b_{4} false(g, ε false)

g

is necessary.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

Baik

Gekhtman

Hamenstädt

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For hyperbolic manifolds of dimension n = 3, this bound is roughly sharp: White [20] proved that for fixed r 2, δ > 0 there exists a constant a = a(r, δ) > 0 such that…”

Section: Introductionmentioning

confidence: 99%