Restriction of toral eigenfunctions to hypersurfaces
aa r X i v : . [ m a t h . SP ] S e p RESTRICTION OF TORAL EIGENFUNCTIONS TOHYPERSURFACES
JEAN BOURGAIN AND ZE´EV RUDNICK
Abstract.
Let T d = R d / Z d be the d -dimensional flat torus. We es-tablish for d = 2 , Introduction and statements
Let M be a smooth Riemannian surface without boundary, ∆ the cor-responding Laplace-Beltrami operator and Σ a smooth curve in M . Burq,G´erard and Tzvetkov [1] established bounds for the L -norm of the restric-tion of eigenfunctions of ∆ to the curve Σ, showing that if − ∆ ϕ λ = λ ϕ λ , λ >
0, then(1) || ϕ λ || L (Σ) ≪ λ / || ϕ λ || L ( M ) and if Σ has non-vanishing geodesic curvature then (1) may be improved to(2) || ϕ λ || L (Σ) ≪ λ / || ϕ λ || L ( M ) Both (1), (2) are saturated for the sphere S .In [1] it is observed that for the flat torus M = T , (1) can be improvedto(3) || ϕ λ || L (Σ) ≪ λ ǫ || ϕ λ || L ( M ) , ∀ ǫ > λ ǫ canbe replaced by a constant, that is whether there is a uniform L restrictionbound. As pointed out by Sarnak [8], if we take Σ to be a geodesic segmenton the torus, this particular problem is essentially equivalent to the currentlyopen question of whether on the circle | x | = λ , the number of lattice pointson an arc of size λ / admits a uniform bound.In [1] results similar to (1) are also established in the higher dimensionalcase for restrictions of eigenfunctions to smooth submanifolds, in particular(1) holds for codimension-one submanifolds (hypersurfaces) and is sharpfor the sphere S d − . Moreover, (2) remains valid for hypersurfaces withnonvanishing curvature [6]. Date : June 19, 2018.
In this note we pursue the improvements of (2) for the standard flat d -dimensional tori T d = R d / Z d , considering the restriction to (codimension-one) hypersurfaces Σ with non-vanishing curvature. Theorem 1.1.
Let d = 2 , and let Σ ⊂ T d be a smooth hypersurface withnon-zero curvature. There are constants < c < C < ∞ and Λ > , alldepending on Σ , so that all eigenfunctions ϕ λ of the Laplacian on T d with λ > Λ satisfy (4) c || ϕ λ || ≤ || ϕ λ || L (Σ) ≤ C || ϕ λ || Observe that for the lower bound, the curvature assumption is necessary,since the eigenfunctions ϕ ( x ) = sin(2 πn x ) all vanish on the hypersurface x = 0. In fact this lower bound implies that a curved hypersurface can-not be contained in the nodal set of eigenfunctions with arbitrarily largeeigenvalues.The proof of Theorem 1.1 (which will be sketched in the next sectionfor the easy case of d = 2) permits also to introduce a notion of “relativequantum limit” for restrictions to Σ as above, but we will not discuss thisfurther here.It is reasonable to believe that Theorem 1.1 holds in any dimension, andone could further conjecture an upper bound without curvature assumptions.At this point, we may only state an improvement of the exponent 1 / Theorem 1.2.
For all d ≥ there is ρ ( d ) < so that if ϕ λ is an eigenfunc-tion of the Laplacian on T d , and Σ ⊂ T d is a smooth compact hypersurfacewith positive curvature, then (5) || ϕ λ || L (Σ) ≪ λ ρ ( d ) || ϕ || Proof of Theorem 1.1 for d = 2Denote by σ the normalized arc-length measure on the curve Σ. Using themethod of stationary phase, one sees that if Σ has non-vanishing curvaturethen the Fourier transform b σ decays as(6) | b σ ( ξ ) | ≪ | ξ | − / , ξ = 0Moreover | b σ ( ξ ) | ≤ b σ (0) = 1 with equality only for ξ = 0, hence(7) sup = ξ ∈ Z | b σ ( ξ ) | ≤ − δ for some δ = δ Σ > T is a trigonometric polynomial ofthe form(8) ϕ ( x ) = X | n | = λ b ϕ ( n ) e ( n · x )(where e ( z ) := e πiz ), all of whose frequencies lie in the set E := Z ∩ λS .As is well known, in dimension d = 2, E ≪ λ ǫ for all ǫ >
0. Moreover,
ESTRICTION OF TORAL EIGENFUNCTIONS TO HYPERSURFACES 3 by a result of Jarnik [7], any arc on λS of length at most cλ / containsat most two lattice points (Cilleruelo and Cordoba [3] showed that for any δ < , arcs of length λ δ contain at most M ( δ ) lattice points and in [4] it isconjectured that this remains true for any δ < E = a α E α where E α ≤ E α , E β ) > cλ / for α = β . Correspondingly wemay write(10) ϕ = X α ϕ α , ϕ α ( x ) = X n ∈E α b ϕ ( n ) e ( nx )so that || ϕ || = P α || ϕ α || and(11) Z Σ | ϕ | dσ = X α X β Z Σ ϕ α ϕ β dσ Applying (6) we see that R Σ ϕ α ϕ β dσ ≪ λ − / if α = β and because E ≪ λ ǫ the total sum of these nondiagonal terms is bounded by λ − / ǫ || ϕ || . Itsuffices then to show that the diagonal terms satisfy(12) δ || φ α || ≤ Z Σ | φ α | dσ ≤ || φ α || This is clear if E α = { n } while if E α = { m, n } then(13) Z Σ | φ α | dσ = | b ϕ ( m ) | + | b ϕ ( n ) | + 2Re b ϕ ( m ) b ϕ ( n ) b σ ( m − n )and then (12) follows from (7). Thus we get Theorem 1.1 for d = 2.3. The higher-dimensional case
The proof of Theorem 1.1 for dimension d = 3 is considerably more in-volved. Arguing along the lines of the two-dimensional case gives an upperbound of λ ǫ . To get the uniform bound of Theorem 1.1 for d = 3 andthe results of Theorem 1.2, we need to replace the upper bound (6) for theFourier transform of the hypersurface measure by an asymptotic expansion,and then exploit cancellation in the resulting exponential sums over thesphere. A key ingredient there is controlling the number of lattice points inspherical caps.3.1. Distribution of lattice points on spheres.
To state some relevantresults, denote as before by E = Z d ∩ λS d − the set of lattice points on thesphere of radius λ . We have E ≪ λ d − ǫ . Let F d ( λ, r ) be the maximalnumber of lattice points in the intersection of E with a spherical cap ofsize r >
1. A higher-dimensional analogue of Jarnik’s theorem implies thatif r ≪ λ / ( d +1) then all lattice points in such a cap are co-planar, hence F d ( r, λ ) ≪ r d − ǫ in that case, for any ǫ >
0. For larger caps, we show:
JEAN BOURGAIN AND ZE´EV RUDNICK
Proposition 3.1. i) Let d = 3 . Then for any η < , (14) F ( λ, r ) ≪ λ ǫ (cid:16) r ( rλ ) η + 1 (cid:17) ii) Let d = 4 . Then (15) F ( λ, r ) ≪ λ ǫ (cid:18) r λ + r / (cid:19) iii) For d ≥ we have (16) F d ( λ, r ) ≪ λ ǫ (cid:18) r d − λ + r d − (cid:19) (the factor λ ǫ is redundant for large d ). The term r d − /λ concerns the equidistribution of E , while the term r d − measures deviations related to accumulation in lower dimensional strata.The second result expresses a mean-equidistribution property of E . Parti-tion the sphere λS into sets C α of size λ / , for instance by intersecting withcubes of that size. Since E ≪ λ ǫ , one may expect that C α ∩ E ≪ λ ǫ .Using Siegel’s mass formula for the number of representations of an integralquadratic form by the genus of a quadratic form, we show (in joint workwith P. Sarnak [2]) that this holds in the mean square: Proposition 3.2. (17) X α [ E ∩ C α )] ≪ λ ǫ , ∀ ǫ > Exponential sums on the sphere.
Let 1 < r < λ and let C , C ′ bespherical r -caps on λS d − of mutual distance at least 10 r . Following theargument for d = 2, we need to bound exponential sums of the form(18) X n ∈ C X n ′ ∈ C ′ b ϕ ( n ) b ϕ ( n ′ ) e ( h ( n − n ′ )) , || ϕ || = 1where h is the support function of the hyper-surface Σ, which appears inthe asymptotic expansion of the Fourier transform of the surface measureon Σ, see [5]. For instance, in the case that Σ = {| x | = 1 } is the unit spherethen h ( ξ ) = | ξ | .Consider from now on the case d = 3. For r < λ − ǫ we simply estimate(18) by F ( λ, r ) (see (14)). When λ − ǫ < r < λ this bound does not sufficesand we need to exploit cancellation in the sum (18). Proposition 3.3.
There is δ > so that (18) admits a bound of λ − δ for λ ≫ . This statement depends essentially on the equidistribution of E in √ λ -caps, as expressed in Proposition 3.2.We finally formulate an example of a bilinear estimate involved in ana-lyzing (18). ESTRICTION OF TORAL EIGENFUNCTIONS TO HYPERSURFACES 5
Proposition 3.4.
Let β ≫ and X, Y ⊂ [0 , arbitrary discrete sets suchthat | x − x ′ | , | y − y ′ | > β − / for x = x ′ ∈ X and y = y ′ ∈ Y . Then (19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ X X y ∈ Y e ( βxy + β / x y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ β / ǫ for all ǫ > . Note that the nonlinear term in the phase function is crucial for a non-trivial bound to hold in this setting.3.3.
Acknowledgements and grant support.
We thank Peter Sarnakfor several stimulating discussions. Supported in part by N.S.F. grant DMS0808042 (J.B.) and Israel Science Foundation grant No. 925/06 (Z.R.).
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E-mail address : [email protected] Raymond and Beverly Sackler School of Mathematical Sciences, Tel AvivUniversity, Tel Aviv 69978, Israel and School of Mathematics, Institute forAdvanced Study, Princeton, NJ 08540
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