  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.
