Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

@article{Wyman2021ImprovedGP,
  title={Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature},
  author={Emmett L. Wyman and Yakun Xi},
  journal={Forum Mathematicum},
  year={2021},
  volume={33},
  pages={789 - 807}
}
Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions eλe_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O⁢((log⁡λ)-12)O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ→∞\lambda\to\infty if 0<|ν|λ<1-δ0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0<δ<10<\delta<1. Our result implies, for instance, that the… Expand
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References

SHOWING 1-10 OF 19 REFERENCES
Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature
We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficient of eigenfunctions $e_\lambda$ over a period geodesic $\gamma$ goesExpand
Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss-Bonnet Theorem
We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctionsExpand
Explicit Bounds on Integrals of Eigenfunctions Over Curves in Surfaces of Nonpositive Curvature
  • E. Wyman
  • Mathematics
  • The Journal of Geometric Analysis
  • 2019
Let ( M ,  g ) be a compact Riemannian surface with nonpositive sectional curvature and let $$\gamma $$ γ be a closed geodesic in M . And let $$e_\lambda $$ e λ be an $$L^2$$ L 2 -normalizedExpand
Quantum Ergodic Restriction Theorems: Manifolds Without Boundary
We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if $${H \subset M}$$ is a smooth hypersurface satisfying a generic microlocal asymmetry condition, thenExpand
On integrals of eigenfunctions over geodesics
If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negativeExpand
Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces
  • Yakun Xi
  • Physics, Mathematics
  • The Journal of Geometric Analysis
  • 2018
We show that for a smooth closed curve $$\gamma $$γ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $$e_\lambda $$eλ and $$e_\mu $$eμ restricted to $$\gammaExpand
On the growth of eigenfunction averages: Microlocalization and geometry
Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2\Delta_g\phi_h=\phi_h$. Given a smooth submanifold $H \subset M$ ofExpand
A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces
We consider periods along closed geodesics and along geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann sur- face. We obtain uniform bounds for suchExpand
Riemannian Geometry
THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss'sExpand
Local Analysis of Selberg's Trace Formula
Preliminaries.- Decompositions of G.- Integral representations of eigenfunctions.- Fourier coefficients and Kloosterman sums.- Computation of some integrals I.- Poincare series and their FourierExpand
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