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

Authors

Christopher D. Sogge (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Yakun Xi (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Cheng Zhang (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

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.

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Published 28 March 2017