Global Harmonic Analysis and the Concentration of Eigenfunctions, Part II:

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1 Global Harmonic Analysis and the Concentration of Eigenfunctions, Part II: Toponogov s theorem and improved Kakeya-Nikodym estimates for eigenfunctions on manifolds of nonpositive curvature Christopher D. Sogge (Johns Hopkins University)

2 Setting and general problem Compact boundaryless manifold (M, g) of dimension n 2. Eigenfunctions: e j (x) = λ 2 j e j (x), e j 2 dv = 1 M Question: How can you detect and measure various types of concentration of eigenfunctions (or, more generally, quasi-modes)? Expect that extreme concentration might occur at certain points and near certain periodic geodesics. We shall concentrate on latter. Can measure concentration in various ways...

3 Extreme behavior on round spheres, S n Consider the standard sphere S n = {x R n+1 : x x x 2 n+1 = 1} Eigenvalues of S n are λ = λ k = k(k + n 1) k, repeating with highest possible multiplicity (very non-generic). d k k n 1 Eigenfunctions are spherical harmonics, restrictions of homogeneous harmonic polynomials in R n+1 to S n.

4 Extreme concentration along periodic geodesics Highest weight spherical harmonics, Q k (x) k n 1 4 (x1 + ix 2 ) k have extreme concentration near equator (periodic geodesic) γ = {x S n : 0 = x = (x 3,..., x n+1 )}. Simplest example of Gaussian beams, Q k (x) = k n 1 k 4 e 2 ln(1 x 2) k n 1 4 e k 2 d(x,γ)2 k n T (γ), k 2 1 where T k 1 2 (γ) denotes a k 1 2 tubular neighborhood about γ. Since equator has codimension n 1 conclude Q k L p (S n ) k n 1 4 {x S n : d(x, γ) k } p k n 1 2 ( p ), p 2 (1)

5 Spherical harmonics Visual representation of first few spherical harmonics. Blue portions represent regions where function is positive, and yellow where it is negative. Distance of surface from origin represents size in angular direction (θ, φ) coordinates on S 2. (Wikepedia)

6 Saturation of L p -norms of eigenfunctions In 1988 showed that e λ L p (M) λ n 1 2 ( p ) e λ L 2 (M), 2 < p 2(n+1) n 1, (2) which are saturated by the Q k due to (1). Also obtained sharp estimates for p > 2(n+1) n 1, which are saturated on S n by zonal functions (concentrating at points). There the exponent is n( p ) 1 2. Knowing when you can improve these estimates for large exponents has been much better understood than the case where 2 < p < 2(n+1) n 1, and the ones for the critical exponent p c = 2(n+1) n 1 have only just been obtained. The case of improved estimates for p > p c was studied by Bérard 1970s (nonpositive curvature) and CS-Zelditch (a necessary and sufficient condition on improvement).

7 Related problems 1: Lower bounds for L 1 norms In work on nodal sets, CS-Zelditch 2011 showed that one has the universal lower bounds λ n 1 4 e λ L 1 (M). (3) This follows from Hölder s inequality and above L p estimates (2): 1 = e λ 2 e λ θp 1 e λ 1 θp p ( e λ θp n 1 1 λ 2 ( )) 1 θ p p, θ p = p 2(p 1). By this argument, any improvement on (2) leads to improved lower bounds for e λ 1. By the above, no improvement is possible on S n because of the Q λ. Recently showed that if (3) is saturated then, like for the Q λ, must be a unit length geodesic γ Π and a λ 1 2 tube, T λ 1 (γ), about it so that 2 {x Tλ 1 (γ) : e λ(x) cλ n } n 1 Tλ 2 1 (γ) = λ 2.

8 Nodal set bounds and lower bounds for L 1 norms Given a real-valued e λ, consider its nodal set N λ = {x M : e λ (x) = 0}, and let N λ denote its (n 1)-dimensional Hausdorff measure. Following earlier work of CS-Zelditch 2011, Hezari-CS 2012 showed that λ ( e λ dv ) 2 Nλ. (4) M Plugging in the lower bound (3) we obtain the lower bound of Colding-Minicozzi 2011: n 1 1 λ 2 N λ. Conjecture of Yau says that N λ λ, and this was verified in real analytic case by Donnelly-Fefferman c Results of C-M and independently of CS-Zelditch were first to get better than exponentially decaying lower bounds.

9 Related problems 2: Kakeya-Nikodym norms We ve seen that the Q λ have most of their mass in T λ 2 1 (γ) if γ is the equator. We have also seen that if the L 1 lower bound (3) is saturated then there must be a unit length geodesic γ Π so that λ n 1 4 e λ (x) on much of T λ 1 (γ). 2 In both of these cases, have that the Kakeya-Nikodym norms satisfy ( ) 1 e λ KN = sup e λ 2 2 dv γ Π T λ 1 (γ) 2 1. Problem: One trivially has e λ KN 1 and so one wonders on which (M, g) can one have improvements e λ KN = o(1), as λ?

10 Kakeya-Nikodym norms and L p, p < p c norms for surfaces Theorem (Bourgain 2009, CS 2011) If n = 2 the following are equivalent: 1 e λ 4 = o(λ 1 8 ) 2 sup γ Π γ e λ 2 ds = o(λ 1 2 ) 3 e λ KN = o(1). Remarks By interpolation, 1) yields improved L p bounds for all 2 < p < p c = 6 2) Represents an improvement over restriction estimates of Burq-Gérard-Tzvetkov 2007, γ e λ 2 ds = O(λ 1 2 ), which is another estimate saturated by the Q λ CS introduced the KN-norms (essential in higher dimensions). Bourgain showed 1) = 2). Trivially 2) = 3), and by Hölder 1) = 3). CS showed that 3) = 1).

11 Shrinking geodesic tubes and KN-norms e λ 2 KN = sup e λ (x) 2 dv. γ Π T λ 1 (γ) 2

12 Extensions to higher dimensions Blair and CS 2015 showed that for higher dimensions as well, e λ KN = o(1) if and only if e λ L p (M) = o(λ n 1 2 ( p ), 2 < p < p c : Theorem If 2 < p < 2(n+1) n 1 then there is a θ p,n > 0 so that e λ L p (M) λ n 1 2 ( p ) e λ θp,n KN e λ 1 θp,n L 2 (M). (5) In 2014 for n = 2 CS-Zelditch and 2015 for n > 2 Blair-CS showed that e λ KN = o(1) if (M, g) has nonpositive sectional curvatures and so: Corollary If (M, g) has nonpositive sectional curvatures e λ L p (M) = o(λ n 1 2 ( p ) ). Leads to improved lower bounds for N λ under this curvature assumption.

13 Improved Kakeya-Nikodym estimates Although we were able to prove o(1) KN estimates and hence o-l p estimates, p < p c, there was no rate of improvement until recently: Theorem (Blair-CS 2015) Suppose that (M, g) has nonpositive curvature. Let Then (log λ) 1 2, if n = 2 c(λ) = (log λ) 1 log log λ, if n = 3 (log λ) 1, if n 4. sup e λ 2 dv c(λ). (6) γ Π T λ 2 1 (γ) Moreover, if n = 2, sup γ Π γ e λ 2 ds c(λ)λ 1 2.

14 Some corollaries Using the L p vs KN bound (5) get following improvements for L p -norms: Corollary If 2 < p < p c = 2(n+1) n 1 there is a σ p,n > 0 so that e λ L p (M) λ n 1 2 ( p ) (log λ) σp,n. (7) Using the proof of the lower bounds (3) see that this implies λ n 1 4 (log λ) σn e λ 1 and so by (4) get Corollary Let (M, g) be as above, then there is a µ n > 0 so that n 1 1 λ 2 (log λ) µ n N λ. In particular, if n = 3, (log λ) µ N λ for all µ < 16.

15 Improvements for the critical space L p c, p c = 2(n+1) n 1 Using the improved L p, p < p c, bounds (7) along with local estimates and techniques of Bourgain 1991 and Bak and Seeger 2011 (latter two valid w/out curvature assumptions) can also get estimates for critical space: Theorem (2015) Let (M, g) have nonpositive curvature. Then there is a σ n > 0 so that e λ L pc λ 1 pc (log log λ) σn. (8) The universal estimate e λ L pc = O(λ 1 pc ) is saturated both by the highest weight spherical harmonics Q λ (concentrating along periodic geodesics) as well as the zonal functions, Z λ, (concentrating at points) on S n. Besides o-improved L 4 (γ) restriction estimates for n = 2 with Xuehua Chen 2014, the bounds (8) are the only ones that improve such estimates. Would be very interesting to replace log log by log in (8).

16 Ideas in proof of improved KN-estimates: Preliminary reduction Let ρ S(R) satisfy ρ(0) = 1 and ˆρ(t) = 0, t > 1, and let P = g. Then the operator ρ(t (λ P)) satisfies ρ(t (λ P))e µ = ρ(t (λ µ))e µ if e µ is an e.f. w/ e.v. µ. These are reproducing operators for the e λ : ρ(t (λ P))e λ = e λ as ρ(0) = 1. Hence, suffices to show that for T log λ ρ(t (λ P))f L 2 (T λ 2 1 (γ)) (log λ) σ f L 2 (M), (9)

17 The kernel of reproducing operators and universal cover Since ρ(t (λ + P)) has a kernel which is O(λ N ) suffices prove analog of (9) involving operator w/ kernel 1 ˆρ(t/T )e iλt( cos(t g ) ) (x, y)dt. πt By properties of ˆρ integrand vanishes if t > T and operator in integral is solution operator for wave eq. By Cartan-Hadamard theorem if (R n, g) is the universal cover and if Γ α : R n M are the deck transformations (translation by elements of Z n if M = T n ) have the analog of Poisson summation formula if M D (fundamental domain) cos(t g )(x, y) = α Γ( cos(t g ) ) (x, α(y)), x, y D M. (10)

18 Universal cover and fundamental domain D M In model case where M = T n, D = ( π, π] n and the α are translation by elements of Z n. If (M, g) is negatively curved the picture is more complicated:

19 Triangles in negative curvature (skewed) (Wikipedia)

20 Propagation of singularities and Toponogov s theorem Let γ(s), s R, denote the lift of the extension of the center geodesic γ(s) of our tube T λ 1 2 (γ). Using microlocal analysis and propagation of singularities guess that the main contributions to 1 ˆρ(t/T )e iλt( cos(t g ) ) (x, α(y))dt, x T πt λ 1 (γ) 2 α Γ occur when α(y) is a distance 1 from γ. Thus, if true, even though the number of terms in sum grow exponentially in T, the number of these terms grow linearly, and so can estimate them. Could estimate the totality of the O(exp(cT )) trivial terms where dist(y, γ) > C if could show they make an angle λ δ when T log λ. A miracle occurs: THEY DO BECAUSE OF TOPONOGOV!!

21 Toponogov s theorem: Girl watching

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