Focal points and sup-norms of eigenfunctions
|
|
- Philippa Logan
- 6 years ago
- Views:
Transcription
1 Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1 /
2 Background Compact real analytic manifold (M, g) of dimension n. Eigenfunctions: e j (x) = λ j e j (x), e j dv = 1 Give fundamental modes of vibration: u j (t, x) = cos tλ j e j (x). Know e j L (M) Cλ n 1 j Question: When can you improve this estimate for eigenfunctions (or, more generally, quasi-modes)? Since u j (t, x) provide high-frequency solutions of wave equations, ( t )u j = 0, expect answer to depend on long-term dynamics of geodesic flow (known to be linked to propagation of singularities for t ) Chris Sogge Focal points and sup-norms of eigenfunctions /
3 Terminology to be used Say that e j L (M) = O(λ n 1 j ), if, as above e j L (M) Cλ n 1 j. We are interested in when we have improvements, e j L (M) = o(λ n 1 j ), meaning that lim sup λ j λ n 1 j e j L (M) = 0. If we cannot do this, we say that e j L (M) = Ω(λ n 1 j ), meaning that lim sup λ j λ n 1 j e j L (M) > 0. (We ll have to capture Ω(λ n 1 j ) sup-norm behavior using quasi-modes, though, in which case we ll also be able to use liminf over natural sequence µ k of frequencies arising from the geometry) Chris Sogge Focal points and sup-norms of eigenfunctions 3 /
4 Loops through a point Let Φ t (x, ξ) = (x(t), ξ(t)), with (x(0), ξ(0)) = (x, ξ) S x M, denote unit-speed geodesic flow in cosphere bundle. Thus, γ = {x(t)} is geodesic in M starting at x. Let L x0 = {ξ S x 0 M : Φ t (x 0, ξ) = (x 0, ξ(t)), some t > 0} denote initial directions of loops through x 0. Figures from A. Besse, Manifolds all of whose geodesics are closed, (Springer, 1978). Chris Sogge Focal points and sup-norms of eigenfunctions 4 /
5 Why real analytic? In the real analytic case, can show that the set of looping directions satisfies either L x0 = 0 or L x0 = S x 0 M AND there is an l > 0 so that Φ l (x 0, ξ) = (x 0, η(ξ)). (i.e., all geodesics starting at x 0 loop back in time l). Say that x 0 a self-focal point [sfp] in second case Chris Sogge Focal points and sup-norms of eigenfunctions 5 /
6 A Few Examples T n, then L x = 0 for all x. (No spf s) S n, have Φ π (x, ξ) = (x, ξ), i.e., periodic flow (Everything a sfp) Surfaces of revolution. Periodic at poles Triaxial ellipsoid: x + z = 1, 0 < a < b < c < 1. Loops of b c equal length thru 4 umbilic points a + y Latter has much different dynamics: Call these 4 points dissipative. Chris Sogge Focal points and sup-norms of eigenfunctions 6 /
7 Dissipative self-focal points x 0 M If x 0 is a spf, there s a minimum l > 0 (first return time) so that Φ l (x 0, ξ) = (x 0, η(ξ)) ξ Sx 0 M. Call η : Sx 0 M Sx 0 M first return map Defines unitary operator (Perron-Frobenius operator) U = U x0 : L (S x 0 M) L (S x 0 M) f Uf (ξ) = f (η(ξ)) J(ξ), J = Jacobian. Define: sfp x 0 M dissipative if U has no invariant L -functions Chris Sogge Focal points and sup-norms of eigenfunctions 7 /
8 von Neumann Ergodic Theorem and self-focal points Assume x is a self-focal point. Then if 1 1 on S x M and η ν = η η η (ν-times), if J ν (ξ) is Jacobian and dµ is Liouville measure on S x M, by von Neumann, as T, T 1 T ν=1 S x M J ν (ξ) dµ(ξ) = T 1 T U ν 1, 1 Π(1), 1, where Π : L (S x M) L (S x M) is projection onto U-invariant functions. Conclude (as U 0) T 1 T ν=1 S x M J ν (ξ) dµ(ξ) ν=1 { 0, if x dissipative c x > 0, if x not dissipative Chris Sogge Focal points and sup-norms of eigenfunctions 8 /
9 Necessary and sufficient conditions for improved sup-norms Define projection onto δ-bands of spectrum (0 < δ 1) by χ [λ,λ+δ] f = λ j [λ,λ+δ] E j f, where E j f (x) = f, e j e j (x) is proj onto j-th eigenspace. Know χ [λ,λ+1] L (M) L (M) = O(λ n 1 ). Local estimate always sharp. Only possible improvements if use shrinking bands of intervals (i.e., δ = δ(λ) 0 as λ ). Problem: n 1 χ [λ,λ+o(1)] L L = o(λ )? Means: ε > 0, δ(ε) > 0 and Λ ε < so that χ [λ,λ+δ(ε)] f ελ n 1 f, λ Λ ε. (1) Chris Sogge Focal points and sup-norms of eigenfunctions 9 /
10 Main result Theorem Have improved sup-norm bounds, (1), if and only if every self-focal point is dissipative. Since χ [λ,λ+δ] e j = e j if λ j [λ, λ + δ], conclude that we have the following result for eigenfunctions: Corollary If all self-focal points are dissipative then e j = o(λ n 1 j ). Chris Sogge Focal points and sup-norms of eigenfunctions 10 /
11 Some earlier results S-Zelditch (00) Have (1) if L x = 0 for all x M. S-Toth Zelditch (011). Same if R x = 0, where R x S x M denotes the set of recurrent directions for the flow. Specifically initial unit directions ξ such that, given ε > 0, there is a t 0 so that Φ t0 (x, ξ) S x M (loops back) and Φ t0 (x, ξ) (x, ξ) < ε. Note that if C x S x M is the set of initial directions for periodic geodesics (i.e. smoothly closed loops) through x, then have C x R x L x and so L x = 0 = R x = 0 = C x = 0. By the Poincaré recurrence theorem if x is a sfp and R x = 0 then the Perron-Frobenius operator U x cannot have 0 f L (S x M) with U x f = f. For then f dµ a finite invariant measure in the class of dµ. Whence, by Poincaré, a.e. point with respect to f dµ would be recurrent and so R x > 0. Chris Sogge Focal points and sup-norms of eigenfunctions 11 /
12 Some earlier results, continued Recall Weyl formula: N(λ) = c M λ n + O(λ n 1 ), if N(λ) = #λ j λ. By the Duistermaat-Guillemin Theorem (1975), if the set C S M of all periodic points for the geodesic flow is of measure zero, then the error term in the Weyl formula is o(λ n 1 ). Our results imply that if the error term in Weyl law is Ω(λ n 1 ) then not only must one have C > 0, but also there must be a non-dissipative sfp. For many years thought that C x = 0 for all x M, should be sufficient for the o(λ n 1 ) sup-norm bounds (1). WRONG (?) On the other hand, as we ll see below, if there is a non-dissipative sfp x 0 M then the recipe for producing quasi-modes that are large there is related to another theorem in D-G ( 75). Chris Sogge Focal points and sup-norms of eigenfunctions 1 /
13 Equivalent quasi-mode formulation of Main Theorem Functions φ λk C (M) are quasi-modes of order o(λ) ( fat quasi-modes ) if φ λk dv = 1 and ( + λ k )φ λ k = o(λ k ). () M (N.B. as λ j λ k = λ j λ k (λ j + λ k ) measures L -concentration in o(1)-sized intervals) Need an extra condition besides () when n 4. Corollary Have φ λk = o(λ n 1 k ) for every sequence λ k if and only if every self-focal point is dissipative. Chris Sogge Focal points and sup-norms of eigenfunctions 13 /
14 Ω(λ n 1 ) bounds at a non-dissipative sfp x 0 M If x 0 is a non-dissipative sfp, let µ k = π l (k + β/4), k = 1,,..., (3) where β is number of conjugate points (w/ multiplicity) along a unit speed geodesic {γ(t) : 0 t < l}, where l is first return time and γ(0) = γ(l) = x 0 Then if T k (slow enough) the coherent state at x 0 φ µk (y) = µ n 1 k ρ(t k (µ k λ j )) e j (x 0 )e j (y) j=0 satisfies (), and has φ µk (x 0 ) µ n 1 k, assuming that 0 ρ S(R) has ρ(t) 1, t 1, ˆρ 0, and supp ˆρ [ 1, 1]. Chris Sogge Focal points and sup-norms of eigenfunctions 14 /
15 Related earlier result of Duistermaat-Guillemin (1975) Theorem (Duistermaat-Guillemin) If geodesic flow on M is periodic with minimal period l > 0 and µ k as in (3) and intervals I k are given by I k = [µ k Ck 1, µ k + Ck 1 ], with C > 1 sufficiently large but fixed, have #{λ j λ : λ j k=1 I k} 1, as λ. #{λ j λ} Chris Sogge Focal points and sup-norms of eigenfunctions 15 /
16 Ideas in proofs Want to show that if all sfp s are dissipative then, given ε > 0, if T (ε), Λ(ε) large n 1 χ [λ,λ+t 1 (ε)] L L ελ, λ Λ(ε), or this cannot happen if there are non-dissipative sftp s. Concentrate for now on former. By orthogonality, χ [λ,λ+t 1 (ε)] L L = sup x M λ j [λ,λ+t 1 (ε)] ( ej (x) ), and so task is equivalent to showing that ( ej (x) ) ε λ n 1, λ 1 (4) λ j [λ,λ+t 1 (ε)] By earlier results of S-Zelditch, know have uniform such bounds near any point x 0 M with L x0 = 0. So need same near given dissipative sfp. Chris Sogge Focal points and sup-norms of eigenfunctions 16 /
17 o-bounds near dissipative self-focal point x 0 M Use above bump function 0 ρ S satisfying ρ(t) 1, t [ 1, 1] and supp ˆρ [ 1, 1] and ˆρ 0. Have λ j [λ,λ+t 1 (ε)] ( ej (x) ) ρ(t (λ λ j )) (e j (x)), j=0 T = T (ε), and so would have (4) near our diss sfp x 0 if could show there that ρ(t (λ λ j )) (e j (x)) ε λ n 1 (5) j=0 Use ρ(t (λ λ j )) (e j (x)) = 1 ˆρ(t/T )e itλ ( e itλ j (e j (x)) ) dt πt j=0 j = 1 T ˆρ(t/T )e itλ ( e it ) (x, x) dt πt T Chris Sogge Focal points and sup-norms of eigenfunctions 17 /
18 FIOs and propagation of singularities Near any t 0 (mod C ) can write exp( it )(x, y) as finite sum j = 1,..., N(t 0 ) of terms (π) n R n e is j (t,y,ξ) ix ξ a j (t, x, y, ξ) dξ, where a S 0, a(lν, x, x, ξ) = e iνβπ/ J ν (ξ), ν Z, t S j = p(y, x S j ) Factors i νβ called Maslov factors Our assumptions (basically) also imply that exp( it )(y, y) is smooth if y close to sfp x and t supp ˆρ( /T ) is not close to {lν : ν = 0, ±1, ±,... } [ T, T ] Chris Sogge Focal points and sup-norms of eigenfunctions 18 /
19 Punch line via stationary phase Using above facts write ξ = rω, ω Sx M and use stationary phase in t, r variables. Miracle: Modulo lower order terms in λ, LHS of (5) is ρ(t (λ λj ))(e j (x)) = λ n 1 ( e ilνλ e iνβπ/)ˆρ(lν) ( 1 J T ν (ω)dµ(ω) ) (6) Sx M ν T If x = x 0 dissipative sfp then (mod l.o.t. s) this is Cλ n 1 T 1 U ν 1, 1, ν T and coefficient is small if T large by the von Neumann ergodic Theorem If x non-diss sfp and λ = µ k as in (3) then blue factor in (6) is one and ˆρ 0. Thus, if 0 < c 0 = lim T T 1 ν T Uν 1, 1, the LHS of (6) c 0 µ n 1 k for every fixed T > 1 if λ 1 Chris Sogge Focal points and sup-norms of eigenfunctions 19 /
20 Remarks and questions Did not use real analyticity in essential way. For instance, get )-sized quasi-modes at any x M if Φ l (x, ξ) Sx M for some l > 0, f L, Uf = f, f 0, and the measure of initial directions of loops through x shorter than l has measure zero. Ω(µ n 1 k As in the real analytic case above, the q.m. s satisfy φ µk (x) = Ω(µ n 1 k ) and M φ µk dv = 1 and ( + µ k )φ µ k = o(µ k ). Question: Can you reach a stronger conclusion by replacing the red term by O(1)? (More traditional definition of q.m. s.) If the Perron-Frobenius operator Uf (ξ) = J(ξ) f (η(ξ)) had a smooth invariant function, would be able to solve a transport equation and do this. In our results, the conclusion or assumption just concerns L -invariant functions (used to obtain o-bounds via von Neumann when all sfp s diss). Chris Sogge Focal points and sup-norms of eigenfunctions 0 /
21 Remarks and questions, continued By interpolation with CS 88 L p estimates: Know that if all sfp s are dissipative then every quasi-mode as in () must satisfy φ λ L p (M) = o(λ n( 1 1 p ) 1 ), p > (n + 1) n 1 (7) What about endpoint p = (n+1) n 1? Can show that all sfp s being dissipative is a necessary condition for (7) when p = (n+1) n 1. Much recent work (Bourgain, CS, CS-Zelditch, Blair-CS) on obtaining improved L p estimates when < p < (n+1) n 1. Relevant thing for this range: concentration along periodic geodesics (as opposed to concentration at points). Different proofs for this range. Boils down to improving geodesic restriction estimates of Burq-Gérard-Tzvetkov (n = ) and geodesic shrinking tube estimates of X. Chen-CS and M. Blair-CS. Chris Sogge Focal points and sup-norms of eigenfunctions 1 /
22 Some applications Bourgain-Shao-CS-Yao: Improved L p estimates for large p lead to improvements of earlier resolvent estimates of Shen and dos Santos Ferreira-Kenig-Salo. Possible for torus (big improvement) and nonpositive curvature (log-improvements) Improvements for < p < (n+1) n 1 lead to improved known results for nodal sets/domains: Colding-Minicozzi, CS-Zelditch, Blair-CS; Gosh-Reznikov-Sarnak... Chris Sogge Focal points and sup-norms of eigenfunctions /
Focal points and sup-norms of eigenfunctions
Focal points and sup-norms of eigenfunctions Chris Sogge (Johns Hopkins University) Joint work with Steve Zelditch (Northwestern University)) Chris Sogge Focal points and sup-norms of eigenfunctions 1
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part II:
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
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part III:
Global Harmonic Analysis and the Concentration of Eigenfunctions, Part III: Improved eigenfunction estimates for the critical L p -space on manifolds of nonpositive curvature Christopher D. Sogge (Johns
More informationOn L p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds
On p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds Chris Sogge (Johns Hopkins University) Joint work with: Jean Bourgain (IAS) Peng Shao (JHU) Xiaohua Yao (JHU, Huazhong
More informationEigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationYAIZA CANZANI AND BORIS HANIN
C SCALING ASYMPTOTICS FOR THE SPECTRAL PROJECTOR OF THE LAPLACIAN YAIZA CANZANI AND BORIS HANIN Abstract. This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise
More informationEndpoint resolvent estimates for compact Riemannian manifolds
Endpoint resolvent estimates for compact Riemannian manifolds joint work with R. L. Frank to appear in J. Funct. Anal. (arxiv:6.00462) Lukas Schimmer California Institute of Technology 3 February 207 Schimmer
More informationThe Gaussian free field, Gibbs measures and NLS on planar domains
The Gaussian free field, Gibbs measures and on planar domains N. Burq, joint with L. Thomann (Nantes) and N. Tzvetkov (Cergy) Université Paris Sud, Laboratoire de Mathématiques d Orsay, CNRS UMR 8628 LAGA,
More informationVariations on Quantum Ergodic Theorems. Michael Taylor
Notes available on my website, under Downloadable Lecture Notes 8. Seminar talks and AMS talks See also 4. Spectral theory 7. Quantum mechanics connections Basic quantization: a function on phase space
More informationNodal lines of Laplace eigenfunctions
Nodal lines of Laplace eigenfunctions Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada s 60th birthday Friday, August 10, 2007 Steve Zelditch Department of Mathematics
More informationRecent developments in mathematical Quantum Chaos, I
Recent developments in mathematical Quantum Chaos, I Steve Zelditch Johns Hopkins and Northwestern Harvard, November 21, 2009 Quantum chaos of eigenfunction Let {ϕ j } be an orthonormal basis of eigenfunctions
More informationYAKUN XI AND CHENG ZHANG
IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES ON RIEMANNIAN SURFACES WITH NONPOSITIVE CURVATURE YAKUN XI AND CHENG ZHANG Abstract. We show that one can obtain improved L 4 geodesic restriction
More informationQuantum ergodicity. Nalini Anantharaman. 22 août Université de Strasbourg
Quantum ergodicity Nalini Anantharaman Université de Strasbourg 22 août 2016 I. Quantum ergodicity on manifolds. II. QE on (discrete) graphs : regular graphs. III. QE on graphs : other models, perspectives.
More informationFractal Weyl Laws and Wave Decay for General Trapping
Fractal Weyl Laws and Wave Decay for General Trapping Jeffrey Galkowski McGill University July 26, 2017 Joint w/ Semyon Dyatlov The Plan The setting and a brief review of scattering resonances Heuristic
More informationStrichartz estimates for the Schrödinger equation on polygonal domains
estimates for the Schrödinger equation on Joint work with Matt Blair (UNM), G. Austin Ford (Northwestern U) and Sebastian Herr (U Bonn and U Düsseldorf)... With a discussion of previous work with Andrew
More informationMicrolocal limits of plane waves
Microlocal limits of plane waves Semyon Dyatlov University of California, Berkeley July 4, 2012 joint work with Colin Guillarmou (ENS) Semyon Dyatlov (UC Berkeley) Microlocal limits of plane waves July
More informationResearch Statement. Yakun Xi
Research Statement Yakun Xi 1 Introduction My research interests lie in harmonic and geometric analysis, and in particular, the Kakeya- Nikodym family of problems and eigenfunction estimates on compact
More informationStrichartz Estimates in Domains
Department of Mathematics Johns Hopkins University April 15, 2010 Wave equation on Riemannian manifold (M, g) Cauchy problem: 2 t u(t, x) gu(t, x) =0 u(0, x) =f (x), t u(0, x) =g(x) Strichartz estimates:
More informationEigenfunction L p Estimates on Manifolds of Constant Negative Curvature
Eigenfunction L p Estimates on Manifolds of Constant Negative Curvature Melissa Tacy Department of Mathematics Australian National University melissa.tacy@anu.edu.au July 2010 Joint with Andrew Hassell
More informationControl from an Interior Hypersurface
Control from an Interior Hypersurface Matthieu Léautaud École Polytechnique Joint with Jeffrey Galkowski Murramarang, microlocal analysis on the beach March, 23. 2018 Outline General questions Eigenfunctions
More informationAn Improvement on Eigenfunction Restriction Estimates for. Compact Boundaryless Riemannian Manifolds with. Nonpositive Sectional Curvature
An Improvement on Eigenfunction Restriction Estimates for Compact Boundaryless Riemannian Manifolds with Nonpositive Sectional Curvature by Xuehua Chen A dissertation submitted to The Johns Hopkins University
More informationSpectral theory, geometry and dynamical systems
Spectral theory, geometry and dynamical systems Dmitry Jakobson 8th January 2010 M is n-dimensional compact connected manifold, n 2. g is a Riemannian metric on M: for any U, V T x M, their inner product
More informationarxiv: v1 [math.ap] 29 Oct 2013
PARK CITY LECTURES ON EIGENFUNTIONS STEVE ZELDITCH arxiv:1310.7888v1 [math.ap] 29 Oct 2013 Contents 1. Introduction 1 2. Results 6 3. Foundational results on nodal sets 18 4. Lower bounds for H m 1 (N
More informationMicrolocal analysis and inverse problems Lecture 3 : Carleman estimates
Microlocal analysis and inverse problems ecture 3 : Carleman estimates David Dos Santos Ferreira AGA Université de Paris 13 Monday May 16 Instituto de Ciencias Matemáticas, Madrid David Dos Santos Ferreira
More informationNew Proof of Hörmander multiplier Theorem on compact manifolds without boundary
New Proof of Hörmander multiplier Theorem on compact manifolds without boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D, 21218, USA xxu@math.jhu.edu Abstract On compact
More informationEstimates on Neumann eigenfunctions at the boundary, and the Method of Particular Solutions" for computing them
Estimates on Neumann eigenfunctions at the boundary, and the Method of Particular Solutions" for computing them Department of Mathematics Australian National University Dartmouth, July 2010 Outline Introduction
More informationDispersive Equations and Hyperbolic Orbits
Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University Outline 1 Introduction 3 Applications 2 Main
More informationThe Schrödinger propagator for scattering metrics
The Schrödinger propagator for scattering metrics Andrew Hassell (Australian National University) joint work with Jared Wunsch (Northwestern) MSRI, May 5-9, 2003 http://arxiv.org/math.ap/0301341 1 Schrödinger
More informationTHE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS
THE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS. STATEMENT Let (X, µ, A) be a probability space, and let T : X X be an ergodic measure-preserving transformation. Given a measurable map A : X GL(d, R),
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationGradient estimates for eigenfunctions on compact Riemannian manifolds with boundary
Gradient estimates for eigenfunctions on compact Riemannian manifolds with boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D 21218 Abstract The purpose of this paper is
More informationResearch Statement. Xiangjin Xu. 1. My thesis work
Research Statement Xiangjin Xu My main research interest is twofold. First I am interested in Harmonic Analysis on manifolds. More precisely, in my thesis, I studied the L estimates and gradient estimates
More informationUnique equilibrium states for geodesic flows in nonpositive curvature
Unique equilibrium states for geodesic flows in nonpositive curvature Todd Fisher Department of Mathematics Brigham Young University Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism
More informationLocal smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping
Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping H. Christianson partly joint work with J. Wunsch (Northwestern) Department of Mathematics University of North
More informationQuantum Ergodicity for a Class of Mixed Systems
Quantum Ergodicity for a Class of Mixed Systems Jeffrey Galkowski University of California, Berkeley February 13, 2013 General Setup Let (M, g) be a smooth compact manifold with or without boundary. We
More informationOn the definitions of Sobolev and BV spaces into singular spaces and the trace problem
On the definitions of Sobolev and BV spaces into singular spaces and the trace problem David CHIRON Laboratoire J.A. DIEUDONNE, Université de Nice - Sophia Antipolis, Parc Valrose, 68 Nice Cedex 2, France
More informationEstimates from below for the spectral function and for the remainder in Weyl s law
for the spectral function and for the remainder in Weyl s law D. Jakobson (McGill), jakobson@math.mcgill.ca I. Polterovich (Univ. de Montreal), iossif@dms.umontreal.ca J. (McGill), jtoth@math.mcgill.ca
More informationarxiv: v2 [math.ap] 11 Jul 2018
LOGARITHMIC IMPROVEMENTS IN L p BOUNDS FOR EIGENFUNCTIONS AT THE CRITICAL EXPONENT IN THE PRESENCE OF NONPOSITIVE CURVATURE arxiv:1706.06704v [math.ap] 11 Jul 018 MATTHEW D. BLAIR AND CHRISTOPHER D. SOGGE
More informationRANDOM FIELDS AND GEOMETRY. Robert Adler and Jonathan Taylor
RANDOM FIELDS AND GEOMETRY from the book of the same name by Robert Adler and Jonathan Taylor IE&M, Technion, Israel, Statistics, Stanford, US. ie.technion.ac.il/adler.phtml www-stat.stanford.edu/ jtaylor
More informationIMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES ON RIEMANNIAN MANIFOLDS WITH CONSTANT NEGATIVE CURVATURE. 1. Introduction
IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES ON RIEMANNIAN MANIFOLDS WITH CONSTANT NEGATIVE CURVATURE CHENG ZHANG Abstract. We show that one can obtain logarithmic improvements of L 2 geodesic
More informationON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY. 1. Introduction
ON STRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS IN COMPACT MANIFOLDS WITH BOUNDARY MATTHEW D. BLAIR, HART F. SMITH, AND CHRISTOPHER D. SOGGE 1. Introduction Let (M, g) be a Riemannian manifold of dimension
More informationStrichartz Estimates for the Schrödinger Equation in Exterior Domains
Strichartz Estimates for the Schrödinger Equation in University of New Mexico May 14, 2010 Joint work with: Hart Smith (University of Washington) Christopher Sogge (Johns Hopkins University) The Schrödinger
More informationSobolev resolvent estimates for the Laplace-Beltrami. operator on compact manifolds. Peng Shao
Sobolev resolvent estimates for the Laplace-Beltrami operator on compact manifolds by Peng Shao A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationEigenfunction Estimates on Compact Manifolds with Boundary and Hörmander Multiplier Theorem
Eigenfunction Estimates on Compact anifolds with Boundary and Hörmander ultiplier Theorem by Xiangjin Xu A dissertation submitted to the Johns Hopkins University in conformity with the requirements for
More informationThe quasi-periodic Frenkel-Kontorova model
The quasi-periodic Frenkel-Kontorova model Philippe Thieullen (Bordeaux) joint work with E. Garibaldi (Campinas) et S. Petite (Amiens) Korean-French Conference in Mathematics Pohang, August 24-28,2015
More informationOn L p resolvent and Carleman estimates on compacts manifolds
On L p resolvent and Carleman estimates on compacts manifolds David Dos Santos Ferreira Institut Élie Cartan Université de Lorraine Three days on Analysis and PDEs ICMAT, Tuesday June 3rd Collaborators
More informationProbabilistic Sobolev embeddings, applications to eigenfunctions estimates
Contemporary Mathematics Probabilistic Sobolev embeddings, applications to eigenfunctions estimates Nicolas Bur and Gilles Lebeau Abstract. The purpose of this article is to present some probabilistic
More informationLECTURE 16: CONJUGATE AND CUT POINTS
LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M, g) be Riemannian and γ : [a, b] M a geodesic. Then by definition, exp p ((t a) γ(a)) = γ(t). We know that exp p is a diffeomorphism near
More informationHeat Kernel and Analysis on Manifolds Excerpt with Exercises. Alexander Grigor yan
Heat Kernel and Analysis on Manifolds Excerpt with Exercises Alexander Grigor yan Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany 2000 Mathematics Subject Classification. Primary
More informationA Survey of Inverse Spectral Results
A Survey of Inverse Spectral Results Ivana Alexandrova Riemannian Geometry Spring 2000 The existence of the Laplace-Beltrami operator has allowed mathematicians to carry out Fourier analysis on Riemannian
More informationGradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier Theorem
Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier Theorem Xiangjin Xu Department of athematics, Johns Hopkins University Baltimore, D, 21218, USA Fax:
More informationMinimal submanifolds: old and new
Minimal submanifolds: old and new Richard Schoen Stanford University - Chen-Jung Hsu Lecture 1, Academia Sinica, ROC - December 2, 2013 Plan of Lecture Part 1: Volume, mean curvature, and minimal submanifolds
More informationAverage theorem, Restriction theorem and Strichartz estimates
Average theorem, Restriction theorem and trichartz estimates 2 August 27 Abstract We provide the details of the proof of the average theorem and the restriction theorem. Emphasis has been placed on the
More informationFrom holonomy reductions of Cartan geometries to geometric compactifications
From holonomy reductions of Cartan geometries to geometric compactifications 1 University of Vienna Faculty of Mathematics Berlin, November 11, 2016 1 supported by project P27072 N25 of the Austrian Science
More informationOne-parameter automorphism groups of the injective factor. Yasuyuki Kawahigashi. Department of Mathematics, Faculty of Science
One-parameter automorphism groups of the injective factor of type II 1 with Connes spectrum zero Yasuyuki Kawahigashi Department of Mathematics, Faculty of Science University of Tokyo, Hongo, Tokyo, 113,
More informationarxiv: v1 [math.sp] 12 May 2012
EIGENFUNCTIONS AND NODAL SETS STEVE ZELDITCH arxiv:205.282v [math.sp] 2 May 202 Abstract. This is a survey of recent results on nodal sets of eigenfunctions of the Laplacian on Riemannian manifolds. The
More informationTHE SEMICLASSICAL THEORY OF DISCONTINUOUS SYSTEMS AND RAY-SPLITTING BILLIARDS
THE SEMICLASSICAL THEORY OF DISCONTINUOUS SYSTEMS AND RAY-SPLITTING BILLIARDS DMITRY JAKOBSON, YURI SAFAROV, AND ALEXANDER STROHMAIER Abstract. We analyze the semiclassical limit of spectral theory on
More informationTHE DIRICHLET-JORDAN TEST AND MULTIDIMENSIONAL EXTENSIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 4, Pages 1031 1035 S 0002-9939(00)05658-6 Article electronically published on October 10, 2000 THE DIRICHLET-JORDAN TEST AND MULTIDIMENSIONAL
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More information1. Plurisubharmonic functions and currents The first part of the homework studies some properties of PSH functions.
MATH 263: PROBLEM SET 2: PSH FUNCTIONS, HORMANDER S ESTIMATES AND VANISHING THEOREMS 1. Plurisubharmonic functions and currents The first part of the homework studies some properties of PSH functions.
More informationFOURIER TRANSFORMS OF SURFACE MEASURE ON THE SPHERE MATH 565, FALL 2017
FOURIER TRANSFORMS OF SURFACE MEASURE ON THE SPHERE MATH 565, FALL 17 1. Fourier transform of surface measure on the sphere Recall that the distribution u on R n defined as u, ψ = ψ(x) dσ(x) S n 1 is compactly
More informationSINGULARITIES OF LAGRANGIAN MEAN CURVATURE FLOW: MONOTONE CASE
SINGULARITIES OF LAGRANGIAN MEAN CURVATURE FLOW: MONOTONE CASE ANDRÉ NEVES Abstract. We study the formation of singularities for the mean curvature flow of monotone Lagrangians in C n. More precisely,
More informationEigenfunctions and nodal sets
Surveys in Differential Geometry XVIII Eigenfunctions and nodal sets Steve Zelditch Abstract. This is a survey of recent results on nodal sets of eigenfunctions of the Laplacian on Riemannian manifolds.
More informationA gentle introduction to Quantum Ergodicity
A gentle introduction to Quantum Ergodicity Tuomas Sahlsten University of Bristol, UK 7 March 2017 Arithmetic study group, Durham, UK Joint work with Etienne Le Masson (Bristol) Supported by the EU: Horizon
More informationDECOUPLING INEQUALITIES IN HARMONIC ANALYSIS AND APPLICATIONS
DECOUPLING INEQUALITIES IN HARMONIC ANALYSIS AND APPLICATIONS 1 NOTATION AND STATEMENT S compact smooth hypersurface in R n with positive definite second fundamental form ( Examples: sphere S d 1, truncated
More informationarxiv: v1 [math.ap] 10 Apr 2013
QUASI-STATIC EVOLUTION AND CONGESTED CROWD TRANSPORT DAMON ALEXANDER, INWON KIM, AND YAO YAO arxiv:1304.3072v1 [math.ap] 10 Apr 2013 Abstract. We consider the relationship between Hele-Shaw evolution with
More informationSRB measures for non-uniformly hyperbolic systems
SRB measures for non-uniformly hyperbolic systems Vaughn Climenhaga University of Maryland October 21, 2010 Joint work with Dmitry Dolgopyat and Yakov Pesin 1 and classical results Definition of SRB measure
More informationAlmost sure global existence and scattering for the one-dimensional NLS
Almost sure global existence and scattering for the one-dimensional NLS Nicolas Burq 1 Université Paris-Sud, Université Paris-Saclay, Laboratoire de Mathématiques d Orsay, UMR 8628 du CNRS EPFL oct 20th
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationL19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1
L19: Fredholm theory We want to understand how to make moduli spaces of J-holomorphic curves, and once we have them, how to make them smooth and compute their dimensions. Fix (X, ω, J) and, for definiteness,
More informationChaos, Quantum Mechanics and Number Theory
Chaos, Quantum Mechanics and Number Theory Peter Sarnak Mahler Lectures 2011 Hamiltonian Mechanics (x, ξ) generalized coordinates: x space coordinate, ξ phase coordinate. H(x, ξ), Hamiltonian Hamilton
More informationTHE LONGITUDINAL KAM-COCYCLE OF A MAGNETIC FLOW
THE LONGITUDINAL KAM-COCYCLE OF A MAGNETIC FLOW GABRIEL P. PATERNAIN Abstract. Let M be a closed oriented surface of negative Gaussian curvature and let Ω be a non-exact 2-form. Let λ be a small positive
More information0. Introduction. Suppose that M is a compact Riemannian manifold. The Laplace operator is given in local coordinates by f = g 1/2
ASIAN J. MATH. c 2006 International Press Vol. 10, No. 1, pp. 115 126, March 2006 004 EIGENFUNCTIONS OF THE LAPLACIAN ON COMPACT RIEMANNIAN MANIFOLDS HAROLD DONNELLY Key words. Mathematical physics, quantum
More informationCohomology of the Mumford Quotient
Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten
More informationCHAPTER 8. Smoothing operators
CHAPTER 8 Smoothing operators Lecture 8: 13 October, 2005 Now I am heading towards the Atiyah-Singer index theorem. Most of the results proved in the process untimately reduce to properties of smoothing
More informationMASLOV INDEX. Contents. Drawing from [1]. 1. Outline
MASLOV INDEX J-HOLOMORPHIC CURVE SEMINAR MARCH 2, 2015 MORGAN WEILER 1. Outline 1 Motivation 1 Definition of the Maslov Index 2 Questions Demanding Answers 2 2. The Path of an Orbit 3 Obtaining A 3 Independence
More informationThe topology of positive scalar curvature ICM Section Topology Seoul, August 2014
The topology of positive scalar curvature ICM Section Topology Seoul, August 2014 Thomas Schick Georg-August-Universität Göttingen ICM Seoul, August 2014 All pictures from wikimedia. Scalar curvature My
More informationTHE ASYMPTOTIC MASLOV INDEX AND ITS APPLICATIONS
THE ASYMPTOTIC MASLOV INDEX AND ITS APPLICATIONS GONZALO CONTRERAS, JEAN-MARC GAMBAUDO, RENATO ITURRIAGA, AND GABRIEL P. PATERNAIN Abstract. Let N be a 2n-dimensional manifold equipped with a symplectic
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationSoliton-like Solutions to NLS on Compact Manifolds
Soliton-like Solutions to NLS on Compact Manifolds H. Christianson joint works with J. Marzuola (UNC), and with P. Albin (Jussieu and UIUC), J. Marzuola, and L. Thomann (Nantes) Department of Mathematics
More informationAdapted complex structures and Riemannian homogeneous spaces
ANNALES POLONICI MATHEMATICI LXX (1998) Adapted complex structures and Riemannian homogeneous spaces by Róbert Szőke (Budapest) Abstract. We prove that every compact, normal Riemannian homogeneous manifold
More informationQuantum decay rates in chaotic scattering
Quantum decay rates in chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski (Berkeley) National AMS Meeting, New Orleans, January 2007 A resonant state for the partially open stadium billiard, computed
More informationMath 426H (Differential Geometry) Final Exam April 24, 2006.
Math 426H Differential Geometry Final Exam April 24, 6. 8 8 8 6 1. Let M be a surface and let : [0, 1] M be a smooth loop. Let φ be a 1-form on M. a Suppose φ is exact i.e. φ = df for some f : M R. Show
More informationTHE QUINTIC NONLINEAR SCHRÖDINGER EQUATION ON THREE-DIMENSIONAL ZOLL MANIFOLDS
THE QUINTIC NONLINEAR SCHRÖDINGER EQUATION ON THREE-DIMENSIONAL ZOLL MANIFOLDS SEBASTIAN HERR Abstract. Let (M, g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationCUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION
CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.
More informationLuminy Lecture 2: Spectral rigidity of the ellipse joint work with Hamid Hezari. Luminy Lecture April 12, 2015
Luminy Lecture 2: Spectral rigidity of the ellipse joint work with Hamid Hezari Luminy Lecture April 12, 2015 Isospectral rigidity of the ellipse The purpose of this lecture is to prove that ellipses are
More informationCMS winter meeting 2008, Ottawa. The heat kernel on connected sums
CMS winter meeting 2008, Ottawa The heat kernel on connected sums Laurent Saloff-Coste (Cornell University) December 8 2008 p(t, x, y) = The heat kernel ) 1 x y 2 exp ( (4πt) n/2 4t The heat kernel is:
More informationThe eigenvalue problem in Finsler geometry
The eigenvalue problem in Finsler geometry Qiaoling Xia Abstract. One of the fundamental problems is to study the eigenvalue problem for the differential operator in geometric analysis. In this article,
More informationSEMICLASSICAL LAGRANGIAN DISTRIBUTIONS
SEMICLASSICAL LAGRANGIAN DISTRIBUTIONS SEMYON DYATLOV Abstract. These are (rather sloppy) notes for the talk given in UC Berkeley in March 2012, attempting to explain the global theory of semiclassical
More informationMicrolocal analysis and inverse problems Lecture 4 : Uniqueness results in admissible geometries
Microlocal analysis and inverse problems Lecture 4 : Uniqueness results in admissible geometries David Dos Santos Ferreira LAGA Université de Paris 13 Wednesday May 18 Instituto de Ciencias Matemáticas,
More informationOn a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationTransport Continuity Property
On Riemannian manifolds satisfying the Transport Continuity Property Université de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) I. Statement of the problem Optimal transport on Riemannian
More informationRESTRICTIONS OF THE LAPLACE-BELTRAMI EIGENFUNCTIONS TO SUBMANIFOLDS. N. Burq, P. Gérard & N. Tzvetkov
RESTRICTIONS OF THE LAPLACE-BELTRAMI EIGENFUNCTIONS TO SUBMANIFOLDS by N. Burq, P. Gérard & N. Tzvetkov Abstract. We estimates the L p norm ( p + of the restriction to a curve of the eigenfunctions of
More informationRandom Locations of Periodic Stationary Processes
Random Locations of Periodic Stationary Processes Yi Shen Department of Statistics and Actuarial Science University of Waterloo Joint work with Jie Shen and Ruodu Wang May 3, 2016 The Fields Institute
More informationLecture 1: Derivatives
Lecture 1: Derivatives Steven Hurder University of Illinois at Chicago www.math.uic.edu/ hurder/talks/ Steven Hurder (UIC) Dynamics of Foliations May 3, 2010 1 / 19 Some basic examples Many talks on with
More informationGradient Estimates and Sobolev Inequality
Gradient Estimates and Sobolev Inequality Jiaping Wang University of Minnesota ( Joint work with Linfeng Zhou) Conference on Geometric Analysis in honor of Peter Li University of California, Irvine January
More informationLaplace transforms of L 2 ball, Comparison Theorems and Integrated Brownian motions. Jan Hannig
Laplace transforms of L 2 ball, Comparison Theorems and Integrated Brownian motions. Jan Hannig Department of Statistics, Colorado State University Email: hannig@stat.colostate.edu Joint work with F. Gao,
More information