Sogge, Christopher D.; Xi, Yakun; Zhang, Cheng Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss-Bonnet theorem. (English) Zbl 1377.35204 Camb. J. Math. 5, No. 1, 123-151 (2017). Let \(\mathcal{M}=(M,g)\) be a compact Riemann surface of negative Gaussian curvature. The authors use the Gauss-Bonnet theorem and the Rauch-Toponogov triangle comparison theorems to show that period integrals of eigen functions \(e_\lambda\) of the Laplacian go to zero at a rate of \(O((\log(\lambda)^{-1/2}\;)\). More generally, a similar result pertains provided that the integrals of the Gaussian curvature over an arbitrary geodesic ball of radius \(r\leq1\) are pinched above by \(-\delta r^k\) for some integer \(k\) and for some \(\delta>0\). Section 1 provides an introduction to the subject. Section 2 deals with Hadamard’s theorem. Section 3 introduces the requisite geometric tools; the Gauss-Bonnet theorem plays an essential role. Section 4 deals with stationary phase bounds, Section 5 with kernel bounds, and Section 6 with period integral estimates. Reviewer: Peter B. Gilkey (Eugene) Cited in 1 ReviewCited in 13 Documents MSC: 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35L20 Initial-boundary value problems for second-order hyperbolic equations 58J50 Spectral problems; spectral geometry; scattering theory on manifolds Keywords:eigenfunction; negative curvature; Gauss-Bonnet theorem; Hadamard theorem PDFBibTeX XMLCite \textit{C. D. Sogge} et al., Camb. J. Math. 5, No. 1, 123--151 (2017; Zbl 1377.35204) Full Text: DOI arXiv