On Eigenvalues of Geometrically Finite Hyperbolic Manifolds with - - PowerPoint PPT Presentation

On Eigenvalues of Geometrically Finite Hyperbolic Manifolds with Infinite Volume

Xiaolong Hans Han

University of Illinois at Urbana-Champaign

April 4, 2020

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Spectrum and Hyperbolic Geometry in 2-d

Geometric Information from Spectrum

For a planar polygon Ω, in 1966 Marc Kac proved ∞

k=1 e−λkt ∼

1 2πt area(Ω) − 1 4 √ 2πt length(∂Ω)

Curvature and Topology

McKean and Singer (1967): for closed Riemannian manifold: can also recover (dimension, volume and) total curvature. In 2-d, with Gauss-Bonnet formula, we can recover Euler characteristic.

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Weyl’s Law and Hyperbolic Manifolds

Example (Number of Eigenvalues and Geometry)

Weyl’s Asymptotic Formula: (M, g) a compact Riemannian manifold. N(λ) = {λk : λk < λ} N(λ) ∼ ωn (2π)n Vol(M)λn/2 Good?

For Hyperbolic Manifolds

Hyperbolic manifolds: Hn/Γ, isometries=Mobius transformations. Similar to T2 = R2/Z2. Volume is a geometric and hence topological invariant for finite-volume hyperbolic manifold.

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Spectrum in Hyperbolic Manifolds

Idea: Geometric data − → Data about spectrum

Mckean, 1970

The bottom of the spectrum of Hn is at least (n − 1)2 4

Schoen, Randol, Doziuk 1980s

M finite volume hyperbolic, λ1(M) ≥ c Vol(M)2 Intuition is that, no "dumbbell" phenomenon as shown by Cheeger.

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Components of Hyperbolic Manifolds

Cusps

Upper half-space Model for Hn, horoball and parabolic subgroup. Metric structure and volume form on a cusp: The metric structure on T: E × [0, ∞), E torus. Volume form ω: ω = hdx ∧ dt where h : E × [0, ∞) → (0, ∞) satisfies ∂ ∂t h(x, t) = −(n − 1)h(x, t)

Tubes

Tubular neighborhood of short geodesic, like solid torus.

Ends with Infinite Volume

Funnel region expanding expnonentially.

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Figure 1: Cusp Figure 2: Tube Figure 3: End

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Upper Bound for First Eigenvalue

Hamenstadt, Mckean: Functional Inequality for Cusp

f smooth with compact support on T.

• T

∇f 2 ≥ (n − 1)2 4

• T

f 2

Idea for Proof

Cauchy-Schwarz Inequality and metric properties of cusps.

For Tubes and Ends

Similar Inequalities and Similar Proofs

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Eigenvalues under Decomposition

Hamenstadt, 2018

For every finite volume oriented Riemannian manifold M of dimension n ≥ 3 and curvature κ ∈ [−b2, −1] and for all k ≥ 0, we have λk(M) ≥ min {1

3λk(Mthick), (n−2)2 12

}

Importance

Not too much is known for eigenvalues for non-compact finite-volume hyperbolic manifolds.

Proposition (Hamenstadt)

f : M → R is a smooth, square integrable function with Raleigh quotient R(f ) < (n − 2)2/12. Then

• M f 2 ≥ 1

3

• M

f 2

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Decomposition of Geometrically Finite Hyperbolic Manifold with Infinite Volume

M is a geometrically finite negatively curved manifold of dimension ≥ 3 with pinched curvature and infinite volume.

Figure 4: Geometrically Finite Manifold with Infinite Volume

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Main Theorem and Argument

Han, 2020

Let M be a geometrically finite, oriented hyperbolic manifold M of dimension n ≥ 3 with infinite volume. Then for all k ≥ 0, we have λk(M) ≥ min{1

3λk((

M), (n−2)2

12

} where r ≥ tanh−1 (n−2)2

(n−1)2 and the boundary condition on

M is Neumann.

Proposition (Han)

f : M → R is a smooth, square integrable function with Raleigh quotient R(f ) < (n − 2)2/12. Then

• M f 2 ≥ 1

3

• M

f 2

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