Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 eigenfunctions $e_\lambda$ over geodesics go to zero at the
rate of $O((\log\lambda)^{-1/2})$ if $\lambda$ are their frequencies. As
discussed in \cite{CSPer}, no such result is possible in the constant curvature
case if the curvature is $\ge0$. Notwithstanding, we also show that these
bounds for period integrals are valid provided that integrals of the curvature
over all geodesic balls of radius $r\le 1$ are pinched from above by $-\delta
r^N$ for some fixed $N$ and $\delta>0$. This allows, for instance, the
curvature to be nonpositive and to vanish of finite order at a finite number of
isolated points. Naturally, the above results also hold for the appropriate
type of quasi-modes.

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The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas