Orthonormal Spectral Cluster Bounds on Manifolds with Nonpositive Curvature

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold with nonpositive sectional curvature. We prove sharp, logarithmically improved spectral cluster bounds for orthonormal systems in the supercritical range. More precisely, for spectral windows of size $(\log \lambda {)}^{-1}$, we obtain the orthonormal analogue of the logarithmically improved $L^q$ estimates of Hassell-Tacy. Our argument combines the universal orthonormal spectral cluster bounds of Frank-Sabin with Bérard-type kernel estimates and a generalization of the Bourgain-Shao-Sogge-Yao multiplier estimate to the orthonormal setting.