Period integrals in nonpositively curved manifolds

Let $M$ be a compact Riemannian manifold without boundary. We investigate the integrals of $L^2$-normalized Laplace eigenfunctions over closed submanifolds. General bounds for these quantities were obtained by Zelditch [23], and are sharp in the case where $M$ is the standard sphere. However, as with sup norms of eigenfunctions, there are many interesting settings where improvements can be made to these bounds, e.g. where $M$ is a negatively curved surface and the submanifold is a geodesic (see [6, 18]).

So far, improvements in the nonpositive curvature setting have been confined to the two-dimensional case (see works of Chen and Sogge [6]; Sogge, Xi, and Zhang [18]; and the author [20, 22]). Here, we provide two theorems which extend these results into the higher dimensional setting. First, we provide an improvement of a half power of $\operatorname{log}$ over the standard bounds provided the submanifold has codimension $2$ and $M$ has strictly negative sectional curvature. Second, we provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. We use the usual tools, such as the Hadamard parametrix and the method of stationary phase, but critical to our argument is a computation of the Hessian of the distance function on the universal cover of $M$.

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This article was supported in part by NSF grant DMS-1665373.

Received 19 September 2018

Accepted 7 May 2019

Published 12 January 2021