Period integrals in nonpositively curved manifolds

We provide an improvement of a half power of log to standard bounds on integrals of Laplace eigenfunctions over submanifolds of codimension 2 or more, where the ambient space is a compact Riemannian manifold with negative sectional curvature. We provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. This result extends previous ones obtained in the 2-dimensional setting by Chen and Sogge; Sogge, Xi, and Zhang; and the author.