EXPLICIT BOUNDS ON INTEGRALS OF EIGENFUNCTIONS OVER CURVES IN SURFACES OF NONPOSITIVE CURVATURE

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Date
2018-03-30
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Johns Hopkins University
Abstract
We investigate bounds on integrals of $L^2$-normalized Laplace eigenfunctions over curves in compact surfaces of nonpositive curvature. In general, these integrals are bounded. However, decay can be obtained in certain circumstances, e.g. Sogge, Xi, and Zhang showed that integrals of eigenfunctions over closed geodesics in negatively curved surfaces have are bounded by a constant times $1/\sqrt{\log \lambda}$, where $\lambda$ is the frequency of the eigenfunction. We show this bound holds for integrals of eigenfunctions over a broad class of curves by using a combination of Sogge, Xi, and Zhang's methods and other geometric tools which we develop in section 3. In particular, we obtain the same decay provided the curvature of the curve avoids, pointwise, the curvature of the tangent circles of infinite radius.
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Keywords
Eigenfunctions, compact manifolds, nonpositive curvature
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