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. 
Christopher D. Sogge, Yakun Xi and Cheng Zhang
math.AP math.CA math.DG math.SP