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Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss-Bonnet theorem. (English) Zbl 1377.35204

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.

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
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