Period Integrals in Nonpositively CurvedManifolds

Emmett L. Wyman

Department of MathematicsJohns Hopkins University

Tianyuan Advanced SeminarBeijing, China 2018

Background

(M, g) is an n-dimensional, compact, boundarylessRiemannian manifold.

∆g is the Laplace-Beltrami operator.

ej an orthonormal basis of eigenfunctions.

∆gej = −λ2j ej .〈ej , ek〉 = δjk .

Σ is a d-dimensional, closed, immersed submanifold in M withsurface measure σ.

Question

What does

∫Σej dσ look like as λj →∞?

The Torus

Flat torus T2 = R2/2πZ2.If Σ is a closed geodesic, there is a sequence of exponentialsej with λj →∞ which are constant along Σ.∫

Σej dσ = O(1).

If Σ has nonvanishing curvature,∫Σej dσ = O(λ

−1/2+�j ).

General Bounds

Theorem (Zelditch ’92)

In general,

∑λj≤λ

∣∣∣∣∫Σej dσ

∣∣∣∣2 ∼ λn−d + O(λn−d−1).Implies...∫

Σej dσ = O(λ

n−d−12

j ).

If R(λ)↗∞, there exists a density 1 subsequence with∫Σej dσ = O(R(λ)λ

−d/2j ).

Sharpness on the Sphere

The O(λn−d−1

2j ) bound is always sharp on the sphere S

n.

Sn has spectrum {√

k(k + n − 1) : k = 0, 1, 2, . . .}, and gapsbetween distinct eigenvalues are at least 1.

There exists C so that∑λj∈[λ,λ+C ]

∣∣∣∣∫Σej dσ

∣∣∣∣2 ≥ λn−d−1.In every interval of length C , there are at most C distincteigenvalues, hence one of them must have

∑λj=λ

∣∣∣∣∫Σej dσ

∣∣∣∣2 ≥ 1C λn−d−1j .Then, pick a maximizing eigenfunction.

Hyperbolic Surfaces

The problem of bounding integrals of eigenfunctions over curvesoriginated in the hyperbolic setting.

(Good ’83 and Hejhal ’82) Let Σ be a closed geodesic in acompact hyperbolic surface. Then,∫

Σej dσ = O(1).

(Reznikov ’15) Conjectured O(λ−1/2+�j ) for when Σ is a

geodesic or circle. Unlikely to be proven with the currenttechniques.

Negatively Curved Surfaces

Let M be a compact surface with (nonconstant) negative sectionalcurvature. Chen and Sogge provided the first improvement to thebound.

(Chen and Sogge ’15) If Σ is a closed geodesic,∫Σej dσ = o(1).

(Sogge, Xi, Zhang ’17) The integral is O(1/√

log λj). Alsothe sectional curvature of M is allowed to vanish of finiteorder.

What Curves Get Improved Bounds?

Think propagation of singularities.

(Euclidean) The curvature of geodesics remains 0 when flowedalong the normal geodesics. Not so for curves withnonvanishing curvature.

(Hyperbolic) The curvature of geodesics changes when flowedalong the normal geodesics. Same for circles. Not so forhorocycles.

It looks like circles of infinite radius may not get improved bounds.Other curves might.

Horocycles and a Curvature Definition

Let M be a compact surface with nonpositive sectional curvature.The universal cover M̃ is diffeomorphic to R2 by theCartan-Hadamard theorem. Fix p ∈ M and v ∈ SpM and considerlifts p̃ and ṽ ∈ Sp̃M̃ in the universal cover.

Informal Definition

Let κH(v) denote the limiting curvature of the circle, at p̃, withcenter taken to ∞ along the geodesic ray in direction v .

Facts:

κH(v) is well-defined, continuous, nonnegative.

κH(v) ≡ 0 if K ≡ 0. (Geodesics)κH(v) ≡ 1 if K ≡ −1. (Horocycles)

Main Result

Main Theorem

Let M be a compact surface with nonpositive sectional curvature.Then, ∫

Σej dσ = O(1/

√log λj)

provided κΣ 6= κH(v) for every unit normal vector v to Σ.

In the torus, ‘good curves’ are the ones with nonvanishingcurvature.

In a hyperbolic surface, ‘good curves’ have curvature differentfrom horocycles, e.g. geodesics and circles.

Corollary for Hyperbolic Surfaces

Recall κH(v) ≡ 1 for hyperbolic surfaces. Geodesics have havecurvature 0, and circles have curvature strictly greater than 1.

Corollary

If Σ is a closed geodesic or a circle in a compact hyperbolic surface,∫Σej dσ = O(1/

√log λj).

Is O(1) Sharp for Horocyles?

Question: We know that the O(1) bound is sharp on closedgeodesics in the 2-torus. Is the O(1) bound sharp for horocycles inhyperbolic surfaces?

Answer: Probably not. Canzani, Galkowski, and Toth showed if ejis a QE sequence, then we get o(1) for all curves in M. Hence, notif we assume QUE.

Generalization to Higher Dimensions

Let M have any dimension n. As before, let p ∈ M and v ∈ SpM,and let p̃ ∈ M̃ and ṽ ∈ Sp̃M̃ be lifts to the universal cover.

Informal Definition

Let IIH(v) be the limit of the second fundamental forms of spheres,at p, with centers taken to infinity along the geodesic ray indirection v .

Here the second fundamental form is a vector-valued quadraticform on the tangent space,

IIΣ(X ,Y ) = (∇XY )⊥.

Main Theorems in Higher Dimensions

Theorem (d ≤ n − 2)

If M has strictly negative sectional curvature and d ≤ n − 2,∫Σej dµ = O(λ

n−d−12

j /√

log λj).

Theorem (d = n − 1)

If M has nonpositive sectional curvature and Σ is a closedhypersurface in M such that

rank(IIΣ − IIH(v)) + rank(IIΣ − IIH(−v)) ≥ n

for each v ∈ SNΣ, then∫Σej dσ = O(1/

√log λj).

Corollaries

Some properties of IIH(v):

If −a2 ≥ K ≥ −b2 for some nonnegative constants a and b,

a|X |2 ≤ 〈IIH(v)(X ,X ), v〉 ≤ b|X |2 for X ∈ v⊥ ⊂ TpM.

If S is a sphere of radius r with inward radial vector v ,

0 < 〈IIS(X ,X )− IIH(v)(X ,X ), v〉 ≤1

r|X |2 for X 6= 0

Corollary

We get the O(1/√

log λ) bound if...

Σ is a sphere.

M has negative curvature and Σ is totally geodesic

Sketch of Proof

By a partition of unity, it suffices to show∫Σbej dσ = O(1/

√log λj)

for each b is a smooth function on Σ with small support.

Let χ be nonnegative, χ(0) = 1, and χ̂ supported on [−1, 1].Suffices to show∑

j

χ(T (λj − λ))∣∣∣∣∫

Σbej dσ

∣∣∣∣2 . T−1λn−d−1 + eCTλδ,for some δ < n − d − 1 and where we set T = c log λ.

Sketch of Proof

By using Fourier inversion and writing the half-wave operator kernel

e it√−∆g (x , y) =

∑j

e itλj ej(x)e j(y),

we get

∑j

χ(T (λj − λ))∣∣∣∣∫

Σbej dσ

∣∣∣∣2=

1

2πT

∑j

∫∫∫b(x)b(y)χ̂(t/T )e it(λj−λ)ej(x)ej(y) dt dx dy

=1

2πT

∫∫∫b(x)b(y)χ̂(t/T )e−itλe it

√−∆g (x , y) dt dx dy .

Sketch of Proof

Let β be smooth, b(t) = 1 for |t| ≥ 1, β(t) = 0 for |t| ≥ 2. Fix ageometric constant R independent of T and λ and cut the integralinto β(t/R) and (1− β(t/R)) parts. Set X̂T (t) = β(t/R)χ̂(t/T ).If we reverse our reduction, we get

1

2πT

∫∫∫β(t/R)χ̂(t/T )e−itλe it

√−∆g (x , y)b(x)b(y) dt dx dy

=1

T

∑j

XT (λj − λ)∣∣∣∣∫

Σbej dσ

∣∣∣∣2But,

|XT (λj − λ)| ≤ CN(1 + |λj − λ|)−N T ≥ 1, N = 1, 2, . . .

so this is O(1/T ) by the standard period integral bounds.

Sketch of Proof

It suffices to show

1

2πT

∫∫∫b(x)b(y)(1−β(t/R))χ̂(t/T )e−itλe it

√−∆g (x , y) dt dx dy

= O(eCTλδ).

At the expense of a O(λ−∞) error, we change e it√−∆g to

2 cos(t√−∆g ). Let (M̃, g̃) be the universal cover of M and Γ the

group of deck transformations. Then,

cos(t√−∆g )(x , y) =

∑α∈Γ

cos(t√−∆g̃ )(αx̃ , ỹ).

Sketch of Proof

Hence, it suffices to show∑α∈Γ

∫∫b(x)b(y)K (T , λ;αx̃ , ỹ) dx dy = O(eCTλδ)

where

K (T , λ; x̃ , ỹ) =

∫(1−β(t/R))χ̂(t/T )e−itλ cos(t

√−∆g̃ )(x̃ , ỹ) dt.

Some facts about K (T , λ; x̃ , ỹ):

K (T , λ; x̃ , ỹ) = 0 for dg̃ (x̃ , ỹ) ≥ T . In particular this meansthere are O(eCT ) many nonzero terms in the sum. Suffices togive a bound for each α.

K (T , λ; x̃ , ỹ) = O(λ−∞) for dg̃ (x̃ , ỹ) ≤ R. This is because(1− β(t/R)) cos(t

√−∆g̃ )(x̃ , ỹ) is smooth for dg̃ (x̃ , ỹ) ≤ R.

This lets us ignore α for which dg̃ (αΣ̃, Σ̃) ≤ R.

Sketch of Proof

For dg̃ (x̃ , ỹ) ≥ 1, the Hadamard parametrix lets us write

K (T , λ; x̃ , ỹ) = λn−1

2

∑±

a±(T , λ; x̃ , ỹ)e±iλdg̃ (x̃ ,ỹ) + R(T , λ; x̃ , ỹ)

For each α withR ≤ dg̃ (αΣ̃, Σ̃) ≤ T ,

we must show∣∣∣∣∫∫ b(x)b(y)a±(T , λ;αx̃ , ỹ)e±iλφα(x ,y) dx dy ∣∣∣∣ . eCTλ−d/2if d ≤ n − 2, or . eCTλ−n/2 if d = n − 1, whereφα(x , y) = dg̃ (αx̃ , ỹ).

Sketch of Proof

Suppose x = (x1, . . . , xd) are geodesic normal coordinates of Σ,and similarly for y . Furthermore, suppose (0, 0) is the sole criticalpoint of φα. Then,

The geodesic connecting αx̃ to ỹ at (x , y) = (0, 0) isperpendicular to both αΣ̃ and Σ̃.

∇2x ,yφα(0, 0) =[〈IIΣ − IISỹ (φα), v〉 ∗

∗ 〈IIΣ − IISαx̃ (φα),−v〉

]where the entries in ∗ can be made small by taking R large.When R is very large,

∇2x ,yφα(0, 0) ∼[〈IIΣ − IIH(−v), v〉 0

0 〈IIΣ − IIH(v),−v〉

]

Sketch of Proof

If d ≤ n − 2, let the coordinate curves of (x1, . . . , xd) be inprincipal directions at 0 so that 〈IIΣ, v〉 is diagonal at 0 withdiagonal entries κi . For each i = 1, . . . , d , select the coordinate xiif 〈IIΣ(∂i , ∂i ), v〉 ≥ 0, or yi otherwise. By relabeling, suppose we’veselected (x1, . . . , x`, y`+1, . . . , yd). Note

〈IIΣ − IISỹ (φα), v〉 and 〈IIΣ − IISαx̃ (φα),−v〉

are uniformly positive definite in the variables x1, . . . , x` andy`+1, . . . , yd , respectively. Taking R large enough shows

∇2x ,yφα(0, 0) ∼[〈IIΣ − IIH(−v), v〉 0

0 〈IIΣ − IIH(v),−v〉

]is at least rank d . We win by stationary phase.

Sketch of Proof

If d = n − 1, by our hypotheses,

rank〈IIΣ − IIH(−v), v〉+ rank〈IIΣ − IIH(v),−v〉 ≥ n.

Hence,

∇2x ,yφα(0, 0) ∼[〈IIΣ − IIH(−v), v〉 0

0 〈IIΣ − IIH(v),−v〉

]has rank at least n, and we are done by stationary phase.

References

Y. Canzani, J. Galkowski, and J. A. Toth

Averages of eigenfunctions over hypersurfaces.J.A. Commun. Math. Phys. (2018)

X. Chen and C. D. Sogge

On integrals of eigenfunctions over geodesicsProceedings of the American Mathematical Society, 143.1 (2015), 151-161.

A. Good

Local analysis of Selberg’s trace formulaLecture Notes in Mathematics. Springer-Verlag, Berlin, 1983.

D. A. Hejhal

Sur certaines séries de Dirichlet associées aux géodésiques fermées d’une surface de Riemann compacte.C. R. Acad. Sci. Paris Sér. I Math., 1982.

A. Reznikov

A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces.Forum Math., 2015.

C. D. Sogge, Y. Xi, C. Zhang

Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss Bonnet theoremCamb. J. Math. 5 (2017), no. 1, 123-151.

S. Zelditch

Kuznecov sum formulae and Szegö limit formulae on manifoldsComm. Partial Differential Equations 17 (1992), no. 1-2, 221260.

Appendix