Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces

  • Yakun XiEmail author


We show that for a smooth closed curve \(\gamma \) on a compact Riemannian surface without boundary, the inner product of two eigenfunctions \(e_\lambda \) and \(e_\mu \) restricted to \(\gamma \), \(|\int e_\lambda \overline{e_\mu }\,\text {d}s|\), is bounded by \(\min \{\lambda ^\frac{1}{2},\mu ^\frac{1}{2}\}\). Furthermore, given \(0<c<1\), if \(0<\mu <c\lambda \), we prove that \(\int e_\lambda \overline{e_\mu }\,\text {d}s=O(\mu ^\frac{1}{4})\), which is sharp on the sphere \(S^2\). These bounds unify the period integral estimates and the \(L^2\)-restriction estimates in an explicit way. Using a similar argument, we also show that the \(\nu \)th order Fourier coefficient of \(e_\lambda \) over \(\gamma \) is uniformly bounded if \(0<\nu <c\lambda \), which generalizes a result of Reznikov for compact hyperbolic surfaces, and is sharp on both \(S^2\) and the flat torus \(\mathbb T^2\). Moreover, we show that the analogs of our results also hold in higher dimensions for the inner product of eigenfunctions over hypersurfaces.


Eigenfunction estimates Generalized periods Compact Riemannian surface 

Mathematics Subject Classification

35P20 58J51 42B37 



The author would like to thank Professor Christopher Sogge, Allan Greenleaf and Alex Iosevich for their constant support and mentoring. In particular, the author want to thank Professor Sogge for many constructive comments, and it is a pleasure for the author to thank Professor Greenleaf for many helpful conversations, and for suggesting a related problem. The author also wants to thank Professor Xiaolong Han for some helpful suggestions.


  1. 1.
    Bérard, P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blair, M.D.: On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature. Isr. J. Math. 224(1), 407–436 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blair, M.D., Sogge, C.D.: Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions (2015). arXiv:1510.07726 Google Scholar
  4. 4.
    Blair, M.D., Sogge, C.D.: Refined and microlocal Kakeya–Nikodym bounds for eigenfunctions in two dimensions. Anal. PDE 8(3), 747–764 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bourgain, J.: Geodesic restrictions and $L^{p}$-estimates for eigenfunctions of Riemannian surfaces. In: Linear and Complex Analysis. American Mathematical Society Translation Series 2, vol. 226, pp. 27–35. American Mathematical Society, Providence (2009)Google Scholar
  6. 6.
    Burq, N., Gérard, P., Tzvetkov, N.: Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. Invent. Math. 159(1), 187–223 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burq, N., Gérard, P., Tzvetkov, N.: Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds. Duke Math. J. 138(3), 445–486 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Canzani, Y., Galkowski, J.: On the growth of eigenfunction averages: microlocalization and geometry. Preprint (2017)Google Scholar
  9. 9.
    Canzani, Y., Galkowski, J., Toth, J.A.: Averages of eigenfunctions over hypersurfaces. Preprint (2017)Google Scholar
  10. 10.
    Chen, X.: An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature. Trans. Am. Math. Soc. 367, 4019–4039 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, X., Sogge, C.D.: A few endpoint geodesic restriction estimates for eigenfunctions. Commun. Math. Phys. 329(2), 435–459 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, X., Sogge, C.D.: On integrals of eigenfunctions over geodesics. Proc. Am. Math. Soc. 143(1), 151–161 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Christianson, H., Hassell, A., Toth, J.A.: Exterior mass estimates and ${L}^2$-restriction bounds for neumann data along hypersurfaces. Int. Math. Res. Not. 6, 1638–1665 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Good, A.: Local Analysis of Selberg’s Trace Formula. Lecture Notes in Mathematics, vol. 1040. Springer, Berlin (1983)CrossRefGoogle Scholar
  15. 15.
    Greenleaf, A., Seeger, A.: Fourier integral operators with fold singularities. J. Reine Angew. Math. 455, 35–56 (1994)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guo, Z., Han, X., Tacy, M.: ${L}^p$ bilinear quasimode estimates. Preprint (2015)Google Scholar
  17. 17.
    Hejhal, D.A.: Sur certaines séries de Dirichlet associées aux géodésiques fermées d’une surface de Riemann compacte. C. R. Acad. Sci. Paris Sér. I Math. 294(8), 273–276 (1982)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hezari, H.: Quantum ergodicity and ${L}^p$ norms of restrictions of eigenfunctions. Preprint (2016)Google Scholar
  19. 19.
    Koch, H., Tataru, D., Zworski, M.: Semiclassical ${L}^p$ estimates. Ann. Henri Poincaré 8, 885–916 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Miao, C., Sogge, C.D., Xi, Y., Yang, J.: Bilinear KakeyaNikodym averages of eigenfunctions on compact Riemannian surfaces. J. Funct. Anal. 271, 2752–2775 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mockenhaupt, G., Seeger, A., Sogge, C.D.: Local smoothing of Fourier integral operators and Carleson–Sjölin estimates. J. Am. Math. Soc. 6(1), 65–130 (1993)zbMATHGoogle Scholar
  22. 22.
    Pitt, N.J.E.: A sum formula for a pair of closed geodesics on a hyperbolic surface. Duke Math. J. 143(3), 407–435 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Reznikov, A.: A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces. Forum Math. 27(3), 1569–1590 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sogge, C.D.: Concerning the $L^{p}$ norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal. 77(1), 123–138 (1988)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sogge, C.D.: Kakeya–Nikodym averages and $L^{p}$-norms of eigenfunctions. Tohoku Math. J. (2) 63(4), 519–538 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sogge, C.D.: Fourier Integrals in Classical Analysis. Cambridge Tracts in Mathematics, vol. 210, 2nd edn. Cambridge University Press, Cambridge (2017)Google Scholar
  27. 27.
    Sogge, C.D., Zelditch, S.: On eigenfunction restriction estimates and $L^{4}$-bounds for compact surfaces with nonpositive curvature. In: Advances in Analysis: The Legacy of Elias M. Stein. Princeton Mathematical Series, vol. 50, pp. 447–461. Princeton University Press, Princeton (2014)Google Scholar
  28. 28.
    Sogge, C.D., Xi, Y., Zhang, C.: Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss–Bonnet theorem. Camb. J. Math. 5(1), 123–151 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tataru, D.: On the regularity of boundary traces for the wave equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV. Ser. 26(1), 185–206 (1998)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wyman, E.: Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature. Preprint (2017)Google Scholar
  31. 31.
    Wyman, E.: Integrals of eigenfunctions over curves in surfaces of nonpositive curvature. Preprint (2017)Google Scholar
  32. 32.
    Wyman, E.: Looping directions and integrals of eigenfunctions over submanifolds. Preprint (2017)Google Scholar
  33. 33.
    Xi, Y.: Improved generalized periods estimates on Riemannian surfaces with nonpositive curvature. Preprint (2017)Google Scholar
  34. 34.
    Xi, Y., Zhang, C.: Improved critical eigenfunction restriction estimates on Riemannian surfaces with nonpositive curvature. Commun. Math. Phys. 350, 1299–1325 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zelditch, S.: Kuznecov sum formulae and Szegő limit formulae on manifolds. Commun. Partial Differ. Equ. 17(1–2), 221–260 (1992)zbMATHGoogle Scholar

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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RochesterRochesterUSA

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