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# Cambridge Journal of Mathematics

## Volume 5 (2017)

### Number 1

### Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss–Bonnet Theorem

Pages: 123 – 151

DOI: https://dx.doi.org/10.4310/CJM.2017.v5.n1.a2

#### Authors

#### Abstract

We use the Gauss–Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemannian 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 “On integrals of eigenfunctions over geodesics” [X. Chen and C. D. Sogge, *Proc. Amer. Math. Soc.* **143** (2015), no. 1, 151–161], no such result is possible in the constant curvature case if the curvature is $\geq 0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r \leq 1$ are pinched from above by ${- \delta r}^N$ for some fixed $N$ and $\delta \gt 0$. 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.

#### Keywords

eigenfunction, negative curvature

#### 2010 Mathematics Subject Classification

Primary 35F99. Secondary 35L20, 42C99.

Published 28 March 2017