Spectral projection estimates restricted to uniformly embedded submanifolds

TL;DR

This paper provides new spectral projection estimates for manifolds with nonpositive curvature and bounded geometry, extending previous results to noncompact cases and specific geometric settings.

Contribution

It generalizes spectral projection estimates to noncompact manifolds and establishes sharp bounds on asymptotically hyperbolic surfaces with curvature constraints.

Findings

Estimated $L^2(M) o L^q()$ norms of spectral projections.

Extended results of X. Chen to noncompact manifolds.

Proved sharp spectral estimates on asymptotically hyperbolic surfaces.

Abstract

Let $M$ be a manifold with nonpositive sectional curvature and bounded geometry, and let $\Sigma$ be a uniformly embedded submanifold of $M.$ We estimate the $L^2(M)\to L^q(\Sigma)$ norm of a $\log$-scale spectral projection operator. It is a generalization of result of X. Chen to noncompact cases. We also prove sharp spectral projection estimates of spectral windows of any small size restricted to nontrapped geodesics on even asymptotically hyperbolic surfaces with bounded geometry and curvature pinched below 0.