Integrals of eigenfunctions over curves in surfaces of nonpositive curvature
aa r X i v : . [ m a t h . A P ] A p r INTEGRALS OF EIGENFUNCTIONS OVER CURVES IN SURFACESOF NONPOSITIVE CURVATURE
EMMETT L. WYMAN
Abstract.
Let (
M, g ) be a compact, 2-dimensional Riemannian manifold with non-positive sectional curvature. Let ∆ g be the Laplace-Beltrami operator correspondingto the metric g on M , and let e λ be L -normalized eigenfunctions of ∆ g with eigen-value λ , i.e. − ∆ g e λ = λ e λ . We prove (cid:12)(cid:12)(cid:12)(cid:12)Z R b ( t ) e λ ( γ ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) = o (1) as λ → ∞ where b is a smooth, compactly supported function on R and γ is a curve parametrizedby arc-length whose geodesic curvature κ ( γ ( t )) avoids two critical curvatures k ( γ ′⊥ ( t ))and k ( − γ ′⊥ ( t )) for each t ∈ supp b . k ( v ) denotes the curvature of a circle with centertaken to infinity along the geodesic ray in direction − v .Chen and Sogge prove in [2] the same decay for geodesics in M with strictlynegative curvature. After performing a standard reduction, they lift the relevantquantity to the universal cover and then use the Hadamard parametrix to reduce theproblem to bounding a sum of oscillatory integrals with a geometric phase functions.They use the Gauss-Bonnet theorem to obtain bounds on the Hessian of these phasefunctions and conclude their argument with stationary phase. Our argument followstheirs, except we prove and use properties of the curvature of geodesic circles toobtain bounds on the Hessian of the phase functions. Statement of results.
Let (
M, g ) be a 2-dimensional compact Riemannian manifold. We denote by e λ an L -normalized eigenfunction of the Laplace-Beltrami operator ∆ g on M , i.e. − ∆ g e λ = λ e λ and k e λ k L ( M ) = 1. We are interested in restrictions of eigenfunctions to curves in M ,in particular with the integral(1.1) Z b ( t ) e λ ( γ ( t )) dt where b is a smooth, compactly supported function on R and γ is a smooth unit-speedcurve in M . In the setting that M is a hyperbolic surface and γ is a closed geodesic,Good [4] and Hejhal [5] showed that Z γ e λ dt = O (1) . Later Reznikov [7] demonstrated the same bound can be achieved if γ is allowed to bea circle in M . For M of arbitrary dimension, Zelditch [12] shows, among other things,period integrals over submanifolds of codimension k are O ( λ k − ), implying the O (1)bound above. In the setting where M has negative sectional curvature, Chen and Sogge [2] obtaineddecay(1.2) Z b ( s ) e λ ( γ ( s )) ds = o (1)where γ is a geodesic in M . Moreover, they showed that decay cannot be guaranteed if M is replace with a sphere or a torus, demonstrating the necessity of negative sectionalcurvature. In the case of the sphere, the bound is saturated by the zonal functions alongthe equator. In the case of the torus, for any closed geodesic γ there exists a sequenceof eigenfunctions which are uniformly constant on γ . Sogge, Xi, and Zhang [10] laterimproved this result by slightly weakening the hypotheses on the curvature of M andobtaining an explicit decay of O ((log λ ) − / ).Our main result builds on the work of Chen and Sogge [2] and shows that their bound(1.2) holds for integrals over γ belonging to a wider class of curves. Notation.
For a 2-dimensional Riemannian manifold M , we let K ( p ) denote the sec-tional curvature of M at a point p ∈ M . Let γ be a regular parametrized curve in M .We let κ γ ( t ) denote the geodesic curvature of γ at t , κ γ ( t ) = 1 | γ ′ ( t ) | (cid:12)(cid:12)(cid:12)(cid:12) Ddt γ ′ ( t ) | γ ′ ( t ) | (cid:12)(cid:12)(cid:12)(cid:12) , where D/dt denotes the covariant derivative in the variable t . For any point p ∈ M and v ∈ T p M , we let v ⊥ denote a choice of vector in T p M such that | v ⊥ | = | v | and h v, v ⊥ i = 0. SM = { v ∈ T M : | v | = 1 } denotes the unit sphere bundle over M .Essential to our result is a particular function k on the unit sphere bundle SM , definedbelow. Definition 1.1.
Let (
M, g ) be a 2-dimensional Riemannian manifold, without boundary,with non-positive sectional curvature. Let v ∈ SM and ζ be the geodesic with ζ ′ (0) = v ,and J be a Jacobi field along ζ satisfying(1.3) | J (0) | = 1 and h J (0) , ζ ′ (0) i = 0 . We denote by k ( v ) the unique value such that(1.4) | J ( r ) | = O (1) for r ≤ J satisfies the additional initial condition(1.5) Ddr J (0) = k ( v ) J (0) . We verify that k is well-defined, continuous, and non-negative in Proposition 4.1. Thegeometric meaning of k is clearer after pulling it back to the universal cover of M . Bythe theorem of Hadamard, we identify the universal cover of ( M, g ) with ( R , ˜ g ), where ˜ g is the pullback of g through the covering map. If v and ζ are as in the definition and ˜ ζ isa lift of ζ to R , then k ( v ) denotes the limiting curvature of a circle at ˜ ζ (0) with centerat ˜ ζ ( − R ) as R → ∞ . This fact comes out in the proof of Proposition 4.1 and Remark4.2. Our main result is as follows. Theorem 1.2.
Let ( M, g ) be a compact -dimensional Riemannian manifold withoutboundary and with nonpositive sectional curvature. Let b be a smooth function on R withcompact support and γ be a smooth unit-speed curve satisfying (1.6) κ γ ( t ) = k ( γ ′⊥ ( t )) and κ γ ( t ) = k ( − γ ′⊥ ( t )) for all t ∈ supp b. NTEGRALS OF EIGENFUNCTIONS OVER CURVES 3
Then, (1.7) Z b ( t ) e λ ( γ ( t )) dt = o (1) as λ → ∞ . If M = R / π Z is the flat torus, one can check directly from the definition that k ≡ γ must have nonvanishing curvature by (1.6). In fact, much stronger decay canbe obtained on the torus in this situation. We write e λ ( x ) = X | m | = λ a ( m ) e ix · m where m ∈ Z and X | m | = λ | a ( m ) | = 1 . Hence by Cauchy-Schwarz (cid:12)(cid:12)(cid:12)(cid:12)Z b ( t ) e λ ( γ ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ { m ∈ Z : | m | = λ } / sup | m | = λ (cid:12)(cid:12)(cid:12)(cid:12)Z b ( t ) e iγ ( t ) · m dt (cid:12)(cid:12)(cid:12)(cid:12) . Since γ has nonvanishing curvature, an elementary stationary phase argument tells us thesupremum in the line above is O ( λ − / ). Bounds on the divisor function in the Gaussianintegers give us { m ∈ Z : | m | = λ } = O ( λ ε )for any fixed ε >
0. Hence, we obtain O ( λ − / ε ) decay for (1.7) for the torus. Thisresult is essentially sharp as demonstrated by taking γ to be a circle and b ≡ M is a compact hyperbolic surface, i.e. M has constantsectional curvature −
1. Then, k ≡
1. The hypotheses (1.6) then exclude curves thatlift to horocycles in the universal cover. As in [6], the characters used in the Fouriertransform on the hyperbolic plane are constant on families of horocycles. The authorwould be interested to know of an example of a compact hyperbolic surface and γ withcurvature 1 such that the integral of eigenfunctions over γ saturate the O (1) bound, i.e.lim sup λ →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z b ( t ) e λ ( γ ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) > . To prove our main result, we follow Chen and Sogge’s strategy exactly as in [2]. First,we make a reduction using the Cauchy-Schwarz inequality to phrase the bound in (1.7)as a kernel bound. Second, we lift the problem to the universal cover where we will usea lemma from [2] to write the kernel as a sum of oscillatory integrals. In [2], Chen andSogge use the Gauss-Bonnet theorem to obtain bounds on the derivatives of the phasefunction and conclude their argument with stationary phase. We obtain bounds on thederivatives of the phase function by exploiting our hypotheses on γ and the behavior ofthe curvature of large circles in the universal cover. Acknowledgements.
The author would like to thank his advisor, Christopher Sogge,for providing the initial problem, related materials, feedback, and support . The authorwould also like to thank Yakun Xi and Cheng Zhang for their feedback. This work is partially supported by the NSF.
EMMETT L. WYMAN Standard reduction and lift to the universal cover.
We use Chen and Sogge’s argument in [2] to reduce the bound in (1.7) to two stationaryphase arguments, Propositions 2.2 and 2.3, which we prove using the tools developed inthe previous section.Let ρ ∈ C ∞ ( R ) be a smooth function satisfying ρ (0) = 1 and supp ˆ ρ ⊂ [ − / , / T >
1, we define the operator ρ ( T ( p − ∆ g − λ )) using the spectral theorem, i.e. ρ ( T ( p − ∆ g − λ )) f = X j ρ ( T ( λ j − λ )) E j f where E j is the orthogonal projection of f onto the space spanned by e j . To proveTheorem 1.2, it suffices to show(2.1) (cid:12)(cid:12)(cid:12)(cid:12)Z b ( t ) ρ ( T ( p − ∆ g − λ )) f ( γ ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( CT − + C T λ − / ) / k f k L ( M ) . Where C is a fixed constant and C T is some constant depending on T . Using ρ ( T ( p − ∆ g − λ )) f ( x ) = Z M X j ρ ( T ( λ j − λ )) e j ( x ) e j ( y ) f ( y ) dV ( y ) , Cauchy-Schwarz, and orthogonality , we write the integral in (2.1) as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M Z X j b ( t ) ρ ( T ( λ j − λ )) e j ( γ ( t )) e j ( y ) f ( y ) dt dV ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X j b ( t ) ρ ( T ( λ j − λ )) e j ( γ ( t )) e j ( y ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dV ( y ) / k f k L ( M ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z X j b ( s, t ) χ ( T ( λ j − λ )) e j ( γ ( s )) e j ( γ ( t )) ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / k f k L ( M ) where b ( s, t ) = b ( s ) b ( t ) and χ = | ρ | . Note that supp ˆ χ ⊂ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z X j b ( s, t ) χ ( T ( λ j − λ )) e j ( γ ( s )) e j ( γ ( t )) ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CT − + C T λ − / By Fourier inversion and a change of variables, we have X j χ ( T ( λ j − λ )) e j ( x ) e j ( y ) = 12 π Z X j ˆ χ ( τ ) e iτT ( λ j − λ ) e j ( x ) e j ( y ) dτ = 12 πT Z X j ˆ χ ( τ /T ) e iτ ( λ j − λ ) e j ( x ) e j ( y ) dτ = 12 πT Z ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( x, y ) dτ, The Cauchy-Schwarz reduction here occurred earlier in [2] and [10].
NTEGRALS OF EIGENFUNCTIONS OVER CURVES 5 where the last line follows from writing out the kernel of the half-wave operator e iτ √ − ∆ g , e iτ √ − ∆ g ( x, y ) = X j e iτλ j e j ( x ) e j ( y ) . Hence, we write (2.2) as(2.3) (cid:12)(cid:12)(cid:12)(cid:12)Z Z Z b ( s, t ) ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( γ ( s ) , γ ( t )) dτ ds dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C T λ − / . At this point, we let β ∈ C ∞ ( R ) with β ( τ ) = 1 if | τ | ≤ β ( τ ) = 0 if | τ | ≥
4. Byscaling the metric, we can assume the injectivity radius of M is 10 or more, and by apartition of unity, we may restrict the support of b to lie in an interval of length 1. Wewrite Z ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( x, y ) dτ = Z β ( τ ) ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( x, y ) dτ + Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( x, y ) dτ. We claim the contribution of the β part to the integral in (2.3) is O (1). As noted in [2]and [10], by the proof of Lemma 5.1.3 in [8] and the assumption that the injectivity radiusof M is at least 10 we can write this term as Z β ( τ ) ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( x, y ) dτ = λ / X ± a ± ( λ ; d g ( x, y )) e ± iλd g ( x,y ) + O (1)where a ± satisfies bounds(2.4) (cid:12)(cid:12)(cid:12)(cid:12) d j dr j a ± ( λ ; r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C j r − j − / if r ≥ λ − , and(2.5) | a ± ( λ ; r ) | ≤ Cλ / if 0 ≤ r ≤ λ − . Our claim follows if(2.6) λ / Z Z b ( s, t ) a ± ( λ ; d g ( γ ( s ) , γ ( t ))) e ± iλd g ( γ ( s ) ,γ ( t )) ds dt = O (1) . After perhaps further restricting the support of b , we have by the inverse function theorema smooth change of variables ( s, r ) ( s, t ( s, r )) where r = ( d g ( γ ( s ) , γ ( t )) if s ≥ t − d g ( γ ( s ) , γ ( t )) if s ≤ t. We then rewrite the integral in (2.6) as λ / Z Z | r |≤ λ − + Z Z | r | >λ − ! ˜ b ( s, r ) a ± ( λ ; | r | ) e ± iλ | r | ds dr where we use ˜ b ( s, r ) ds dr to denote b ( s, t ) ds dt . The | r | ≤ λ − part is trivially O (1) by(2.5). The | r | > λ − part is also O (1) after integrating by parts once in r and applying(2.4). Hence we have (2.6), and what is left is to show(2.7) (cid:12)(cid:12)(cid:12)(cid:12)Z Z Z b ( s, t )(1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( γ ( s ) , γ ( t )) dτ ds dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C T λ − / . EMMETT L. WYMAN
We will need to lift the computation to the universal cover. Before we do this, wewant to rephrase (2.7) using cos( τ p − ∆ g ) rather than e iτ √ − ∆ g . This will allow us tomake use of Huygen’s principle after we lift to ensure the kernel we obtain is supportedon a neighborhood of the diagonal. Using Euler’s formula, we write Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ e iτ √ − ∆ g ( x, y ) dτ = 2 Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ cos( τ p − ∆ g )( x, y ) dτ − Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ e − iτ √ − ∆ g ( x, y ) dτ. Writing ˆΦ T ( τ ) = (1 − β ( τ )) ˆ χ ( τ /T ), the latter term becomes X j Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτ ( λ j + λ ) e j ( x ) e j ( y ) dτ = 12 π X j Φ T ( λ j + λ ) e j ( x ) e j ( y ) . The contribution from this term to the integral in (2.7) is rapidly decaying in λ , uniformlyin T . Hence, it suffices to show (cid:12)(cid:12)(cid:12)(cid:12)Z Z Z b ( s, t )(1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ cos( τ p − ∆ g )( γ ( s ) , γ ( t )) dτ ds dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C T λ − / (2.8)We are now ready to lift to the universal cover. We identify the universal cover of M with R equipped with the pullback metric ˜ g . Let Γ be the group of deck transformations.Let ˜ f ∈ C ∞ ( R ) and f ( p ) = X α ∈ Γ ˜ f ( α (˜ p )) , where ˜ p is a lift of p through the covering map. Now let ˜ u (˜ p, t ) be the solution to thewave equation (cid:3) ˜ g ˜ u = 0 with initial data u (˜ p,
0) = ˜ f (˜ p ). Let u ( p, t ) = P α ∈ Γ ˜ u ( α (˜ p ) , t ).Observe that u satisfies the wave equation (cid:3) g u = 0 with initial data u ( p,
0) = f ( p ).Hence, we conclude thatcos( τ p − ∆ g ) f ( p ) = X α ∈ Γ cos( τ p − ∆ ˜ g ) ˜ f ( α (˜ p )) , and so we have cos( τ p − ∆ g )( x, y ) = X α ∈ Γ cos( τ p − ∆ ˜ g )( α (˜ x ) , ˜ y ) , where ˜ x and ˜ y are lifts of x and y through the covering map, respectively. Hence, wewrite (2.8) as(2.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X α ∈ Γ Z Z b ( s, t ) K T,λ (˜ γ α ( s ) , ˜ γ ( t )) ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C T λ − / where K T,λ ( x, y ) = Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ cos( τ p − ∆ ˜ g )( x, y ) dτ, where x and y belong to the universal cover. Here ˜ γ is a lift of γ to the universal cover,and ˜ γ α = α ◦ ˜ γ . Now cos( τ p − ∆ ˜ g )( x, y ) is supported on d g ( x, y ) ≤ | τ | by Huygen’sprinciple, and since ˆ χ ( τ /T ) is suppoted on [ − T, T ], we have that K T,λ is supported on d ˜ g ( x, y ) ≤ T . Hence, the sum in (2.9) is finite. In fact, as noted in [2] and [10], the sumhas O ( e CT ) terms by volume comparison. NTEGRALS OF EIGENFUNCTIONS OVER CURVES 7
To proceed, we will need bounds on K T,λ . We will make use of Lemma 2.4 from [2],stated below.
Lemma 2.1 (Chen and Sogge) . We write K T,λ (˜ x, ˜ y ) = λ / w (˜ x, ˜ y ) X ± a ± ( T, λ ; d ˜ g (˜ x, ˜ y )) e ± iλd ˜ g (˜ x, ˜ y ) + R T,λ (˜ x, ˜ y ) where w is a smooth bounded function on R × R and where for each j = 0 , , , . . . thereis a constant C j independent of T, λ ≥ so that (2.10) (cid:12)(cid:12)(cid:12)(cid:12) d j dr j a ± ( T, λ ; r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C j r − / − j , for r ≥ , and for a constant C T independent of γ and λ such that | R T,λ (˜ x, ˜ y ) | ≤ C T λ − . The contribution of the R T,λ to the sum in (2.9) is bounded by C T λ − , better thanrequired. Moreover since β ( τ ) = 1 if | τ | ≤
3, we have (1 − β ( τ )) cos( τ p − ∆ ˜ g )(˜ x, ˜ y ) issmooth if d ˜ g (˜ x, ˜ y ) ≤
1. Hence for d ˜ g (˜ x, ˜ y ) ≤ Z (1 − β ( τ )) ˆ χ ( τ /T ) e − iτλ cos( τ p − ∆ ˜ g )(˜ x, ˜ y ) = O T ( λ − N )for arbitrary N , and hence the contribution of the identity term in (2.9) is triviallybounded by C T λ − / . We now need only show(2.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ / X α ∈ Γ \ I Z Z b ( s, t ) w (˜ γ α ( s ) , ˜ γ ( t )) a ± ( T, λ ; φ α ( s, t )) e ± iλφ α ( s,t ) ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C T λ − / where φ α ( s, t ) = d ˜ g (˜ γ α ( s ) , ˜ γ ( t )). To do so, we will split the sum into two parts and boundthem separately. Fix R to be determined later (in the proof of Proposition 2.3), and set(2.12) A = { α ∈ Γ : φ α ( s, t ) ≤ R for some ( s, t ) ∈ supp b × supp b } . We will show that the contribution of A to the sum in (2.11) is bounded by a constant,and that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ / X Γ \ A Z Z b ( s, t ) w (˜ γ α ( s ) , ˜ γ ( t )) a ± ( T, λ ; φ α ( s, t )) e ± iλφ α ( s,t ) ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C T λ − / . The above bounds follow from the Propositions 2.2 and 2.3 below, respectively, thenfollows (2.11) and hence Theorem 1.2.
Proposition 2.2.
For any fixed α ∈ A \ I , there exists a constant C α such that (2.13) (cid:12)(cid:12)(cid:12)(cid:12)Z Z b ( s, t ) w (˜ γ α ( s ) , ˜ γ ( t )) a ± ( T, λ ; φ α ( s, t )) e ± iλφ α ( s,t ) ds dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α λ − / . Proposition 2.3.
For any fixed α ∈ Γ \ A , there exists a constant C α such that (2.14) (cid:12)(cid:12)(cid:12)(cid:12)Z Z b ( s, t ) w (˜ γ α ( s ) , ˜ γ ( t )) a ± ( T, λ ; φ α ( s, t )) e ± iλφ α ( s,t ) ds dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C α λ − . EMMETT L. WYMAN Phase function bounds.
To prove Propositions 2.2 and 2.3, we will need bounds on the derivatives of the phasefunction φ α for α = I . First, we bound the mixed partial derivative ∂ s ∂ t φ α , and secondcompute ∂ s φ α in terms of the curvature κ γ of γ and the curvature of circles. We will usethese computations later to obtain bounds on the pure second derivatives of φ α .Let F : supp b × supp b × R be the smooth map defined so that r F ( s, t, r ) is theconstant-speed geodesic with F ( s, t,
0) = ˜ γ ( t ) and F ( s, t,
1) = ˜ γ α ( s ). If ∂ s , ∂ t , and ∂ r are the coordinate vector fields living in the domain of F , then the Lie brackets [ ∂ s , ∂ t ],[ ∂ s , ∂ r ] and [ ∂ t , ∂ r ] all vanish. Hence,(3.1) Dds ∂ t F − Ddt ∂ s F = [ ∂ s F, ∂ t F ] = [ F ∗ ∂ s , F ∗ ∂ t ] = F ∗ [ ∂ s , ∂ t ] = 0and(3.2) Dds ∂ r F − Ddr ∂ s F = 0 and Ddt ∂ r F − Ddr ∂ t F = 0similarly (see for example do Carmo [3]). Now, φ α ( s, t ) = Z | ∂ r F ( s, t, r ) | dr, and so φ α ( s, t ) ∂ s φ α ( s, t ) = Z (cid:28) Dds ∂ r F ( s, t, r ) , ∂ r F ( s, t, r ) (cid:29) dr = Z ∂ r h ∂ s F ( s, t, r ) , ∂ r F ( s, t, r ) i dr = h ˜ γ ′ α ( s ) , ∂ r F ( s, t, i (3.3)where the second line follows from (3.2) and the geodesic equation Ddr ∂ r F = 0, and thethird line by the fundamental theorem of calculus. Moreover, since the curves ˜ γ and˜ γ α are disjoint, φ α is nonvanishing. From this we have the following fact (also notedin [2] and [10]): ∂ s φ α ( s, t ) vanishes if and only if ˜ γ α is perpendicular to the geodesicadjoining ˜ γ α ( s ) and ˜ γ ( t ). This works similarly where ∂ t φ α vanishes, and hence thegradient ∇ φ α ( s, t ) vanishes if and only if ˜ γ and ˜ γ α are both perpendicular to the geodesicadjoining ˜ γ α ( s ) and ˜ γ ( t ). We will appeal to this fact without reference.Now we compute the mixed partial derivative ∂ s ∂ t φ α at a critical point. From (3.3)we obtain(3.4) ∂ s φ α ( s, t ) ∂ t φ α ( s, t ) + φ α ( s, t ) ∂ t ∂ s φ α ( s, t ) = (cid:28) ˜ γ ′ α ( s ) , Ddr ∂ t F ( s, t, (cid:29) . From this computation we derive a useful bound.
Lemma 3.1. If ( s, t ) is a critical point of φ α , | ∂ s ∂ t φ α ( s, t ) | ≤ φ − α . Proof.
Since both ∂ s φ α and ∂ t φ α vanish at ( s, t ), we are done if we can show that theright and side of (3.4) is bounded by 1. Since ∂ t F ( s, t,
1) = 0 and ∂ t F ( s, t,
0) = ˜ γ ′ ( t )is perpendicular to ζ , ∂ t F is a perpendicular Jacobi field to r F ( s, t, r ). Hence if r w ( r ) is the vector field along r F ( s, t, r ) obtained by a parallel transport of ˜ γ ′ ( t ),we write ∂ t F ( s, t, r ) = h ( r ) w ( r ) NTEGRALS OF EIGENFUNCTIONS OVER CURVES 9 where h is a smooth function satisfying h ′′ ( r ) + K ( F ( s, t, r )) h ( r ) = 0where K is the sectional curvature of ( R , ˜ g ), with initial conditions h (0) = 1 and h (1) = 0 . Since ∂ t F must have no conjugate points, h vanishes only at 1, and so h is nonnegativeon [0 , K ≤ h ′′ ≥ , h is convex. Hence,0 ≤ h ( r ) ≤ − r for r ∈ [0 , . The above line and the limit definition of the derivative yield the bound0 ≥ h ′ (1) ≥ − . Since
Ddr ∂ t F ( s, t, r ) = h ′ ( r ) w ( r ) , we have (cid:12)(cid:12)(cid:12)(cid:12) Ddr ∂ t F ( s, t, (cid:12)(cid:12)(cid:12)(cid:12) = 1which along with the fact | ˜ γ α | = 1, yields the desired bound. (cid:3) Now we compute ∂ s φ α . Fix t and let r ζ ( s, r ) denote the unit speed geodesic with ζ ( s,
0) = ˜ γ ( t ) and ζ ( s, φ α ( s, t )) = ˜ γ α ( s ). To avoid ambiguity in the notation, we willfix s and let r = φ α ( s , t ), and compute ∂ s φ α ( s , t ). By (3.3), ∂ s φ α ( s, t ) = h ˜ γ ′ α ( s ) , ∂ r ζ ( s, φ α ( s, t )) i . Differentiating in s yields(3.5) ∂ s φ α ( s , t ) = (cid:28) Dds ˜ γ ′ α ( s ) , ∂ r ζ ( s , r ) (cid:29) + * ˜ γ ′ α ( s ) , Dds (cid:12)(cid:12)(cid:12)(cid:12) s = s ∂ r ζ ( s, φ ( s, t )) + . Now
Dds (cid:12)(cid:12)(cid:12)(cid:12) s = s ∂ r ζ ( s, φ ( s, t )) = Dds ∂ r ζ ( s , r ) + ∂ s φ α ( s , t ) Ddr ∂ r ζ ( s , r ) . The latter term on the right vanishes since r ζ ( s , r ) is a geodesic. The curve s ζ ( s, r ) is a geodesic circle of radius r . Hence, ∂ r ζ and ∂ s ζ are perpendicular by Gauss’lemma and | ∂ r ζ | = 1. Hence, there exists a function κ such that(3.6) Dds ∂ r ζ = κ∂ s ζ. In fact, κ ( s , r ) is the geodesic curvature of the circle s ζ ( s, r ) at s = s . Hence, Dds ∂ r ζ ( s , r ) = κ ( s , r ) ∂ s ζ ( s , r ) . and (3.5) becomes(3.7) ∂ s φ α ( s , t ) = (cid:28) Dds ˜ γ ′ α ( s ) , ∂ r ζ ( s , r ) (cid:29) + κ ( s , r ) h ˜ γ ′ α ( s ) , ∂ s ζ ( s , r ) i . Let θ ∈ [0 , π/
2] denotes the angle of intersection between the curve ˜ γ α and the circle s ζ ( s, r ). We have˜ γ ′ α ( s ) = ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) s = s ζ ( s, φ ( s, t )) = ∂ s ζ ( s , r ) + ∂ s φ ( s , t ) ∂ r ζ ( s , r ) , and since ∂ r ζ and ∂ s ζ are perpendicular, h ˜ γ ′ α ( s ) , ∂ s ζ ( s , r ) i = | ∂ s ζ ( s , r ) | = cos ( θ ) . The line above and (3.7) yields(3.8) ∂ s φ α ( s , t ) = cos( θ )( ± κ γ ( s ) + cos( θ ) κ ( s , r )) , where ± matches the sign of h Dds ˜ γ ′ α , ∂ r ζ i .4. Curvature of circles.
Fix t and let ζ and κ be as in (3.6). To apply (3.8) in any useful way, we need toknow something about the function κ ( s, r ), the curvature of a geodesic circle of radius r centered ˜ γ ( t ). Note by the same argument for (3.1), Dds ∂ r ζ = Ddr ∂ s ζ. This and (3.6) yields D dr ∂ s ζ = Ddr ( κ∂ s ζ ) = ∂ r κ∂ s ζ + κ Ddr ∂ s ζ = ( ∂ r κ + κ ) ∂ s ζ. On the other hand since ∂ s ζ is a perpendicular Jacobi field along r ζ ( s, r ), D dr ∂ s ζ = − K∂ s . Putting these together, we obtain a simple equation for κ ,(4.1) ∂ r κ + κ + K = 0 . We want to compare the behaviors of κ and the quantity k , but first we must verifyDefinition 1.1. Proposition 4.1. k ( v ) as given in Definition 1.1 exists and is unique for each v . More-over k is a continuous, non-negative function on the unit sphere bundle SM .Proof. We use the notation in Definition 1.1. We first observe that k ( v ) does not dependon our choice of J (0). The only other Jacobi field satisfying (1.3) and (1.4) is − J , whichwould yield the same value of k ( v ) in (1.5). Granted, this holds if for a fixed choice of J (0), there is exactly one value for Ddr J (0) such that J satisfies (1.4). We prove this now.Since both J (0) and Ddr J (0) are perpendicular to ζ ′ (0), h J ( r ) , ζ ′ ( r ) i = 0 for all r. Hence, if w ( r ) denotes the vector at ζ ( r ) obtained through a parallel transport of J (0)along ζ , we write J ( r ) = h ( r ) w ( r )for some smooth function h satisfying h ′′ + Kh = 0 and(4.2) h (0) = 1 , (4.3)where K = K ( ζ ( r )) is the sectional curvature at ζ ( r ). We have reduced the problem toproving that there exists a unique function h satisfying (4.2) and (4.3) and also(4.4) | h ( r ) | ≤ C for r ≤ , NTEGRALS OF EIGENFUNCTIONS OVER CURVES 11 and then setting k ( v ) = h ′ (0). We begin with uniqueness. Suppose h and h bothsatisfy (4.2), (4.3), and (4.4). Then the difference u = h − h satisfies (4.2) and (4.4)with initial data u (0) = 0. If u ′ (0) >
0, then u ( r ) > r >
0, otherwise we wouldhave a conjugate point. Hence, u ′′ ≥
0, and so u ( r ) ≥ ru ′ (0), a contradiction. We derivea similar contradiction if u ′ (0) < u ′ (0) = 0.To prove existence, we construct a bounded h as a limit. For all s >
0, let h s denotethe unique function satisfying (4.2), (4.3), and h s ( − s ) = 0. We construct as a limit(4.5) h −∞ = lim s →∞ h s = h + Z ∞ ∂ s h s ds which we will show converges uniformly on compact sets. We then have smooth conver-gence by (4.2). Hence, h ∞ satisfies (4.2) and (4.3). It will then be left to show that h ∞ satisfies (4.4). To prove convergence, we first show(4.6) | ∂ s h s ( r ) | ≤ − rs for r ≤ . Now h s may only vanish at − s , otherwise we have conjugate points. Hence, h s > − s, h ′′ s ≥ − s, h s (0) = 1 and h s ( − s ) = 0,0 ≤ h s ( r ) ≤ (cid:16) rs (cid:17) for − s ≤ r ≤ ≤ h ′ s ( − s ) ≤ s by writing h ′ s ( − s ) as a difference quotient and applying the previous inequality. Nowsince h s ( − s ) = 0 for all s > dds h s ( − s ) = − h ′ s ( − s ) + ∂ s h s ( − s ) . Hence,(4.7) 0 ≤ ∂ s h s ( − s ) ≤ s . Now ∂ s h s also satisfies (4.2) with initial data ∂ s h s (0) = 0. Since ∂ s h s ( − s ) >
0, a similarconvexity argument as before yields bounds0 < ∂ s h s ( r ) ≤ − ∂ s h s ( − s ) rs for − s ≤ r < . (4.6) follows from the above inequality and (4.7). The bound (4.6) implies the pointwiseconvergence of the limit (4.5). Moreover if we fix s >
0, for r ∈ [ − s ,
0] we have(4.8) | h ∞ ( r ) − h s ( r ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ s ∂ s h s ( r ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ − Z ∞ s rs ds = − rs . This implies uniform convergence on compact sets. Similarly, h ∞ ( − s ) = Z ∞ s ∂ s h s ( − s ) ds, which together with (4.6) implies(4.9) 0 < h ∞ ( − s ) ≤ s > . which is stronger than (4.4). This completes the proof of existence. To show that k ( v ) is non-negative, we argue that h ′∞ (0) ≤
0. By (4.9), h ∞ ( r ) does notvanish for r >
0, and so h ′′∞ ( r ) ≥ r ≥
0. However if at the same time h ′∞ (0) > h ∞ would certainly be unbounded on [0 , ∞ ). Hence h ′∞ (0) ≤ k is continuous on SM . To do so, we show that k is continuous onevery continuous path t v ( t ) in SM . If r ζ ( t, r ) is the geodesic with ∂ r ζ ( t,
0) = v ( t ),we let h ∞ ( t, r ) and h s ( t, r ) be as constructed above along the geodesic r ζ ( t, r ). Nowin the limit as t →
0, the sectional curvature K ( ζ ( t, r )) converges to K ( ζ (0 , r )) uniformlyfor r in a compact set. Combined with (4.2), we have for any ε > s > δ > | h s ( t, r ) − h s (0 , r ) | < ε − s ≤ r ≤ | t | < δ . Moreover if r lies in some compact set, by (4.8) there exists s > | h ∞ ( t, r ) − h s ( t, r ) | < ε t . Putting these bounds together, we have | h ∞ ( t, r ) − h ∞ (0 , r ) |≤ | h ∞ ( t, r ) − h s ( t, r ) | + | h s ( t, r ) − h s (0 , r ) | + | h s (0 , r ) − h ∞ (0 , r ) | < ε, i.e. h ∞ ( t, r ) → h ∞ (0 , r ) uniformly for r in a compact set. By (4.2), ∂ r h ∞ ( t, r ) → ∂ r h ∞ (0 , r ) uniformly for r in a compact set. Hence, ∂ r h ( t, r ) → ∂ r h (0 , r ) as t →
0, andin particular k ( v ( t )) → k ( v (0)). (cid:3) Remark . Let ˜ k be given by ˜ k (˜ v ) = k ( v ) where ˜ v is a lift of v to S R . Since the coveringmap is a local isomorphism, ˜ k satisfies Definition 1.1 on the manifold ( R , ˜ g ). From nowon we will work exclusively in the universal cover, noting that k in the hypotheses (1.6)can be freely replaced with ˜ k .We can loosen Definition 1.1 a little bit. If v , ζ , and J are as in Definition 1.1 (herewe replace the manifold M in the definition with the universal cover as justified by theabove remark), except that | J (0) | is allowed to take any value except 0, we may write˜ k ( v ) = | Ddr J (0) || J (0) | . Then, ˜ k ( ζ ′ ( r )) = | Ddr J ( r ) || J ( r ) | = h ′ ( r ) h ( r ) for all r ≥ h as in the proof of Proposition 4.1. It follows that(4.10) ddr ˜ k ( ζ ′ ( r )) = h ′′ ( r ) h ( r ) − h ′ ( r ) h ( r ) = − K − ˜ k ( ζ ′ ( r )) , and hence ˜ k ( ζ ′ ( r )) satisfies the same ordinary differential equation (4.1) as κ . As aconsequence, we have the following lemma. Lemma 4.3.
Let r ζ ( r ) be a unit-speed geodesic in ( R , ˜ g ) and κ ( r ) the geodesiccurvature at ζ ( r ) of the circle of radius r with center at ζ (0) . Then, < κ ( r ) − ˜ k ( ∂ r ζ ( r )) ≤ r − , r > . NTEGRALS OF EIGENFUNCTIONS OVER CURVES 13
Proof.
Since both κ and ˜ k satisfy (4.1), the difference κ − ˜ k satisfies(4.11) ∂ r ( κ − ˜ k ) = − ( κ − ˜ k ) . Since κ ( r ) is large for small r , we can easily guarantee that κ ( r ) > ˜ k ( ζ ′ ( r )) for 0 < r ≪ κ and ˜ k are smooth for r >
0, and since κ − ˜ k = 0 is an equilibrium of (4.11), wehave that κ ( r ) − ˜ k ( ζ ′ ( r )) > r > . Hence ∂ r ( κ − ˜ k ) = − κ + ˜ k κ − ˜ k ( κ − ˜ k ) ≤ − ( κ − ˜ k ) , the inequality a consequence of the fact that κ > ˜ k . By comparison, κ ( r ) − ˜ k ( ζ ′ ( r )) ≤ r − , as desired. (cid:3) Conclusion of the proof of Theorem 1.2.
All that is left is to prove Propositions 2.2 and 2.3
Proof of Proposition 2.2.
Since A is fixed and finite, we may restrict the support of b without worrying about doing so uniformly over elements of A . Fix α ∈ A \ I . Let D denote the diagonal of supp b × supp b . We claim that that(5.1) D ⊂ { ∂ t φ α = 0 } ∪ { ∂ s φ α = 0 } ∪ {∇ φ α = 0 } Provided our claim is true, we restrict the support of b by a fine enough partition of unityso that at least one of the conditions ∂ t φ α = 0, ∂ s φ α = 0, or ∇ φ α = 0 holds on all ofsupp b × supp b . In the first case, the proposition follows by stationary phase [8, Theorem1.1.1] in t , and similarly for the second case. In the third case, the proposition follows bynonstationary phase [8, Lemma 0.4.7].Fix s = t ∈ supp b and suppose ∇ φ α ( s , t ) = 0. To prove (5.1), we need only showthat either ∂ s φ α ( s , t ) = 0 or ∂ t φ α ( s , t ) = 0. Let r ζ ( r ) be the constant-speedgeodesic with ζ (0) = ˜ γ ( t ) and ζ (1) = ˜ γ α ( s ). By the computation (3.8), ∂ s φ α ( s , t ) = ± κ γ ( s ) + κ ( s , φ α ( s , t ))where ± agrees with the sign of h ζ ′ (1) , D/ds ˜ γ α ( s ) i . If h ζ ′ (1) , D/ds ˜ γ ′ α ( s ) i ≥
0, we aredone since κ ( s , φ α ( s , t )) is positive. If not, we will prove that h− ζ ′ (0) , D/dt ˜ γ ′ ( t ) i ≥ , which yields ∂ t φ α ( s , t ) > t . Since α is anisometry, (cid:28) − ζ ′ (0) , Ddt ˜ γ ′ ( t ) (cid:29) = − (cid:28) α ∗ ζ ′ (0) , α ∗ Ddt ˜ γ ′ ( t ) (cid:29) = − (cid:28) α ∗ ζ ′ (0) , Dds ˜ γ ′ α ( s ) (cid:29) . We claim that ζ ′ (1) = α ∗ ζ ′ (0) . As noted earlier, ζ is perpendicular to both γ and γ α since ∇ φ α ( s , t ) = 0. Since α ∗ (˜ γ ′ ( t )) = ˜ γ ′ α ( s ), we need only rule out the possibility that − ζ ′ (1) = α ∗ ζ ′ (0). If this were the case, however, ζ (1 /
2) = α ◦ ζ (1 /
2) by uniqueness, contradicting the fact that α is a deck transformation. Hence, − (cid:28) α ∗ ζ ′ (0) , Dds ˜ γ ′ α ( s ) (cid:29) = − (cid:28) ζ ′ (1) , Dds ˜ γ ′ α ( s ) (cid:29) > , as desired. (cid:3) Proof of Proposition 2.3.
By our hypothesis (1.6) on the curvature of γ , and since k iscontinuous, we restrict the support of b so thatinf s,t ∈ supp b | κ γ ( t ) − k ( ± γ ′⊥ ( s )) | > ε for some small ε >
0. Let ζ be defined as in Section 3, that is let r ζ ( s, t, r ) be theunit-speed geodesic with ζ ( s, t,
0) = ˜ γ ( t ) and ζ ( s, t, φ α ( s, t )) = ˜ γ α ( s ). Moreover let κ ( s, t )denote the curvature at ˜ γ α ( s ) of the circle with center ˜ γ ( t ) and radius φ α ( s, t ). We set R = 2 ε − in (2.12). Since R > ε , Lemma 4.3 tells us | κ ( s, t ) − ˜ k ( ζ ′ ( s, t, φ α ( s, t ))) | < ε, and hence(5.2) | κ ( s, t ) − κ γ α ( t ) | > ε for s, t ∈ I . We claim that the determinant of the Hessian of φ α is nonzero at critical points of φ . Itfollows that, for each α ∈ Γ \ A , supp b × supp b contains finitely many stationary pointsof φ α , all of which are non-degenerate. The desired bound follows by stationary phase [8,Theorem 1.1.4].Suppose ∇ φ α ( s, t ) = 0 at some point ( s, t ) ∈ supp b × supp b . By the bound (3.1) andour assertion that R = 2 ε − , we have | ∂ t ∂ s φ α ( s, t ) | ≤ ε/ . Moreover, by (5.2) and the computation (3.8), we have | ∂ s φ α ( s, t ) | ≥ ε and | ∂ t φ α ( s, t ) | ≥ ε. Hence, the determinant of the hessian | det φ ′′ ( s, t ) | is bounded by | det φ ′′ ( s, t ) | = | ( ∂ s φ α ( s, t ))( ∂ t φ α ( s, t )) − ( ∂ s ∂ t φ α ( s, t )) |≥ | ∂ s φ α ( s, t ) || ∂ t φ α ( s, t ) | − | ∂ s ∂ t φ α ( s, t ) | ≥ ε , which proves our claim. (cid:3) References [1] P. H. B´erard,
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