Period integrals in nonpositively curved manifolds

TL;DR

This paper improves bounds on integrals of Laplace eigenfunctions over certain submanifolds in negatively curved manifolds, extending previous 2D results to higher dimensions and specific hypersurfaces.

Contribution

It extends previous 2D bounds to higher-dimensional negatively curved manifolds and hypersurfaces with specific curvature conditions.

Findings

Improved bounds on eigenfunction integrals with a half power of log

Extension of 2D results to higher dimensions

Applicable to hypersurfaces with curvature conditions

Abstract

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.