Annales Henri Poincaré

, 12:1027 | Cite as

On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

Article

Abstract

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.

References

  1. 1.
    Berry M.: Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A 35(13), 3025–3038 (2002)MathSciNetADSCrossRefMATHGoogle Scholar
  2. 2.
    Castrillón López M., Fernández Mateos V., Muñoz Masqué J.: Total curvature of curves in Riemannian manifolds. Differ. Geom. Appl. 28(2), 140–147 (2010)CrossRefMATHGoogle Scholar
  3. 3.
    Cheng S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51(1), 43–55 (1976)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cilleruelo J., Cordoba A.: Trigonometric polynomials and lattice points. Proc. AMS 115(4), 899–905 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cilleruelo, J., Granville, A.: Lattice points on circles, squares in arithmetic progressions and sumsets of squares, vol. 43, pp. 241–262. CRM Proc. LN, AMS (2007)Google Scholar
  6. 6.
    Cilleruelo J., Granville A.: Close lattice points on circles. Can. J. Math. 61(6), 1214–1238 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Donnelly H., Fefferman C.: Nodal sets of eigenfunctions of Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)MathSciNetADSCrossRefMATHGoogle Scholar
  8. 8.
    Fontes-Merz N.: A multidimensional version of Turán’s lemma. J. Approx. Theory 140(1), 27–30 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jarnik V.: Uber die Gitterpunkte auf konvexen Kurven. Math. Z. 24(1), 500–518 (1926)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Milnor J.W.: On the total curvature of knots. Ann. Math. (2) 52, 248–257 (1950)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nazarov, F.: Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz 5(4), 3–66 (1993); translation in St. Petersburg Math. J. 5(4), 663–717 (1994)Google Scholar
  12. 12.
    Ramana D.S.: Arithmetical applications of an identity for the Vandermonde determinant. Acta Arith. 130(4), 351–359 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  13. 13.
    Risler J.: On the curvature of the real Milnor fiber. Bull. Lond. Math. Soc. 35, 445–454 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.School of Mathematics, Institute for Advanced StudyPrincetonUSA
  2. 2.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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