Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\gamma$ goes to 0 at the
rate of $O((\log\lambda)^{-1/2})$, if $0<\nu<c_0\lambda$, given any $0<c_0<1$.
No such result is possible for the sphere $S^2$ or the flat torus $\mathbb
T^2$. Combined with the quantum ergodic restriction result of Toth and
Zelditch, our results imply that for a generic closed geodesic $\gamma$ on a
compact hyperbolic surface, the restriction $e_{\lambda_j}|_\gamma$ of an
orthonormal basis $\{e_{\lambda_j}\}$ has a full density subsequence that goes
to zero weakly in $L^2(\gamma)$. Our proof consists of a further refinement of
a recent paper by Sogge, Xi and Zhang on the geodesic period integrals
($\nu=0$), which featured the Gauss-Bonnet Theorem as a key quantitative tool
to avoid geodesic rectangles on the universal cover of $M$. In contrast, we
shall employ the Gauss-Bonnet Theorem to quantitatively avoid geodesic
parallelograms. The use of Gauss-Bonnet also enables us to weaken our curvature
condition, by allowing the curvature to vanish at an averaged rate of finite
type.

13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg