Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

More documents

Recommendations

Info

(M, g) of maximal growth in L p norms of eigenfunctions We say that (M, g) has “maximal L p eigenfunction growth” if it possesses a sequence ϕ λjk of eigenfunctions which achieve the universal growth bounds. The standard S n has maximal eigenfunction growth. The flat torus does not. Nor do hyperbolic manifolds. How can one characterize (M, g) of “maximal L p eigenfunction growth” ? When does (M, g) possess a sequence of eigenfunctions achieving the maximal sup norm bound ||ϕ λ || L ∞ ≤ Cλ n−1 2 .
First condition for maximal eigenfunction growth Theorem (Sogge-Z) If there exists a sequence {ϕ λjk } of eigenfunctions which achieves (saturates) the bound ||ϕ λ || L ∞ ≤ λ (n−1)/2 then there must exist a point x ∈ M for which the set L x = {ξ ∈ S ∗ x M : ∃T : exp x T ξ = x} (8) of directions of geodesic loops at x has positive surface measure. Here, exp is the exponential map, and the measure |Ω| of a set Ω is the one induced by the metric g x on T ∗ x M. For instance, the poles x N , x S of a surface of revolution (S 2 , g) satisfy |L x | = 2π.
Page 1 and 2: Park City<
Page 3 and 4: L p norms We recall that ∆ϕ λ =
Page 5 and 6: Sogge upper bounds on L p norms of
Page 7 and 8: Extremals The upper bounds are shar
Page 9 and 10: Spherical harmonics of degree k Let
Page 11 and 12: Oscillation of a spherical harmonic
Page 13 and 14: Coherent state Φ x k at x For each
Page 15 and 16: Normalized zonal harmonics On S n ,
Page 17 and 18: Gaussian Beam: maximizes L p norms
Page 19 and 20: Intuitive principle The fact that t
Page 21 and 22: Lagrangian distributions The Lagran
Page 23 and 24: What is the Lagrangian submanifold?
Page 25: Intensity plot of an ergodic eigenf
Page 29 and 30: Special points and submanifolds ◮
Page 31 and 32: Additional necessary conditions For
Page 33 and 34: Pole of a surface of revolution and
Page 35 and 36: Sketch of proof ◮ Suffices to sho
Page 37 and 38: Local Weyl law and L ∞ estimates
Page 39 and 40: Notation for smoothed remainder Let
Page 41 and 42: Many charts, many oscillatory integ
Page 43 and 44: Pointwise Weyl asymptotics: Safarov
Page 45 and 46: Decomposition of R 1 into loops and
Page 47 and 48: Ergodic theory at the self-focal po
Page 49 and 50: Uniform o(λ n−1 ) outside of sma
Page 51: Summing up ◮ No eigenfunction ϕ

Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

Create successful ePaper yourself

Delete template?

Save as template?