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

• DOI: 10.4310/cjm.2017.v5.n1.a2
• Publication Date: 2017-03-28T15:26:43Z
• Authors: Christopher D. Sogge; Yakun Xi; Cheng Zhang
• Keywords: Eigenfunction; Gaussian curvature; Gauss–Bonnet theorem; Total curvature

We use the Gauss-Bonnet theorem and triangle comparison theorems of Rauch Toponogov to show that on compact Riemannian surfaces negative curvature period integrals eigenfunctions e λ over geodesics go zero at rate O((log λ) -1/2 ) if are their frequencies.As discussed in [4], no such result is possible constant case ≥ 0. Notwithstanding, we also these bounds for valid provided all geodesic balls radius r ≤ 1 pinched from above by -δr N some fixed δ > This allows, instance, be nonpositive vanish finite order a number isolated points.Naturally, results hold appropriate type quasi-modes.