American Institute of Mathematical Sciences

Advanced
  2015, 22: 76-86. doi: 10.3934/era.2015.22.76

Fixed frequency eigenfunction immersions and supremum norms of random waves

Yaiza Canzani 1, and  Boris Hanin 2,

1. 

Department of Mathematics, Harvard University, Cambridge, United States
2. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, United States

Received  January 2015 Revised  June 2015 Published  September 2015

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.
Keywords: Immersions by eigenfunctions, supremum norms, random waves..
Mathematics Subject Classification: 35P05 (Pri), 53C42 (Sec.
Citation: Yaiza Canzani, Boris Hanin. Fixed frequency eigenfunction immersions and supremum norms of random waves. Electronic Research Announcements, 2015, 22: 76-86. doi: 10.3934/era.2015.22.76
References:
[1]

R. Adler and J. Taylor, Random Fields and Geometry,, Springer Monographs in Mathematics, (2007).   Google Scholar

[2]

A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series,, Trans. Amer. Math. Soc., 360 (2008), 4425.  doi: 10.1090/S0002-9947-08-04456-5.  Google Scholar

[3]

P. Bartlett, Theoretical statistics,, lecture 14, ().   Google Scholar

[4]

N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications,, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917.   Google Scholar

[5]

Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law,, preprint, ().   Google Scholar

[6]

J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39.  doi: 10.1007/BF01405172.  Google Scholar

[7]

R. Feng and S. Zelditch, Median and mean of the supremum of $L^2$ normalized random holomorphic fields,, J. Func. Anal., 266 (2014), 5085.  doi: 10.1016/j.jfa.2014.02.012.  Google Scholar

[8]

L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193.  doi: 10.1007/BF02391913.  Google Scholar

[9]

M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves,, Ann. of Math. (2), 177 (2013), 699.  doi: 10.4007/annals.2013.177.2.8.  Google Scholar

[10]

F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus,, Ann. Inst. Fourier (Grenoble), 58 (2008), 299.  doi: 10.5802/aif.2351.  Google Scholar

[11]

V. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary,, (Russian) Funksional. Anal. i Prolzhen., 14 (1980), 25.   Google Scholar

[12]

J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold,, Statist. Decisions, 26 (2008), 35.  doi: 10.1524/stnd.2008.0911.  Google Scholar

[13]

L. Nicolaescu, Complexity of random smooth functions on compact manifolds,, Indiana Univ. Math. J., 63 (2014), 1037.  doi: 10.1512/iumj.2014.63.5321.  Google Scholar

[14]

J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres,, Ph.D. Thesis, (2000).   Google Scholar

[15]

B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure,, Asian J. Math., 7 (2003), 627.   Google Scholar

[16]

G. Tian, On a set of polarized Kähler metrics on algebraic manifolds,, J. Differential Geom., 32 (1990), 99.   Google Scholar

[17]

N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem,, in Séminaire Équations aux dérivées partielles, (): 2008.   Google Scholar

[18]

J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere,, Internat. Math. Res. Notices, 7 (1997), 329.  doi: 10.1155/S1073792897000238.  Google Scholar

[19]

S. Zelditch, Real and complex zeros of Riemannian random waves,, in Spectral Analysis in Geometry and Number Theory, (2009), 321.  doi: 10.1090/conm/484/09482.  Google Scholar

[20]

S. Zelditch, Fine structure of Zoll spectra,, J. Func. Anal., 143 (1997), 415.  doi: 10.1006/jfan.1996.2981.  Google Scholar

[21]

S. Zelditch, Szegö kernels and a theorem of Tian,, Internat. Math Res. Notices, (1998), 317.  doi: 10.1155/S107379289800021X.  Google Scholar

show all references

References:
[1]

R. Adler and J. Taylor, Random Fields and Geometry,, Springer Monographs in Mathematics, (2007).   Google Scholar

[2]

A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series,, Trans. Amer. Math. Soc., 360 (2008), 4425.  doi: 10.1090/S0002-9947-08-04456-5.  Google Scholar

[3]

P. Bartlett, Theoretical statistics,, lecture 14, ().   Google Scholar

[4]

N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications,, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917.   Google Scholar

[5]

Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law,, preprint, ().   Google Scholar

[6]

J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39.  doi: 10.1007/BF01405172.  Google Scholar

[7]

R. Feng and S. Zelditch, Median and mean of the supremum of $L^2$ normalized random holomorphic fields,, J. Func. Anal., 266 (2014), 5085.  doi: 10.1016/j.jfa.2014.02.012.  Google Scholar

[8]

L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193.  doi: 10.1007/BF02391913.  Google Scholar

[9]

M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves,, Ann. of Math. (2), 177 (2013), 699.  doi: 10.4007/annals.2013.177.2.8.  Google Scholar

[10]

F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus,, Ann. Inst. Fourier (Grenoble), 58 (2008), 299.  doi: 10.5802/aif.2351.  Google Scholar

[11]

V. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary,, (Russian) Funksional. Anal. i Prolzhen., 14 (1980), 25.   Google Scholar

[12]

J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold,, Statist. Decisions, 26 (2008), 35.  doi: 10.1524/stnd.2008.0911.  Google Scholar

[13]

L. Nicolaescu, Complexity of random smooth functions on compact manifolds,, Indiana Univ. Math. J., 63 (2014), 1037.  doi: 10.1512/iumj.2014.63.5321.  Google Scholar

[14]

J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres,, Ph.D. Thesis, (2000).   Google Scholar

[15]

B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure,, Asian J. Math., 7 (2003), 627.   Google Scholar

[16]

G. Tian, On a set of polarized Kähler metrics on algebraic manifolds,, J. Differential Geom., 32 (1990), 99.   Google Scholar

[17]

N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem,, in Séminaire Équations aux dérivées partielles, (): 2008.   Google Scholar

[18]

J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere,, Internat. Math. Res. Notices, 7 (1997), 329.  doi: 10.1155/S1073792897000238.  Google Scholar

[19]

S. Zelditch, Real and complex zeros of Riemannian random waves,, in Spectral Analysis in Geometry and Number Theory, (2009), 321.  doi: 10.1090/conm/484/09482.  Google Scholar

[20]

S. Zelditch, Fine structure of Zoll spectra,, J. Func. Anal., 143 (1997), 415.  doi: 10.1006/jfan.1996.2981.  Google Scholar

[21]

S. Zelditch, Szegö kernels and a theorem of Tian,, Internat. Math Res. Notices, (1998), 317.  doi: 10.1155/S107379289800021X.  Google Scholar

[1]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
[2]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020215
[3]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[4]

Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[5]

Daniel T. Wise. Nonpositive immersions, sectional curvature, and subgroup properties. Electronic Research Announcements, 2003, 9: 1-9.

[6]

Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
[7]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[8]

Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671
[9]

Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329
[10]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[11]

Giovanni Bellettini, Matteo Novaga, Shokhrukh Yusufovich Kholmatov. Minimizers of anisotropic perimeters with cylindrical norms. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1427-1454. doi: 10.3934/cpaa.2017068
[12]

Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587
[13]

Guojing Zhang, Steve Rosencrans, Xuefeng Wang, Kaijun Zhang. Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1061-1079. doi: 10.3934/dcds.2009.25.1061
[14]

Tohru Wakasa, Shoji Yotsutani. Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2010, 9 (2) : 539-561. doi: 10.3934/cpaa.2010.9.539
[15]

Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks & Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239
[16]

Haïm Brezis. Remarks on some minimization problems associated with BV norms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7013-7029. doi: 10.3934/dcds.2019242
[17]

Dmitry Jakobson and Alexander Strohmaier. High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows. Electronic Research Announcements, 2006, 12: 87-94.

[18]

Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115
[19]

Delfina Gómez, Sergey A. Nazarov, Eugenia Pérez. Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions. Networks & Heterogeneous Media, 2011, 6 (1) : 1-35. doi: 10.3934/nhm.2011.6.1
[20]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences