On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature

TL;DR

This paper improves bounds on the growth of eigenfunction restrictions to geodesics in nonpositively curved manifolds by developing a new formula for the third variation of arc length, sharpening previous results.

Contribution

It introduces a coordinate-free formula for the third variation of arc length, enabling logarithmic improvements in restriction bounds for eigenfunctions.

Findings

Sharper upper bounds on $L^p$ norms of eigenfunction restrictions.

Development of a coordinate-free expression for third variation of arc length.

Logarithmic improvements over previous bounds.

Abstract

We consider upper bounds on the growth of $L^p$ norms of restrictions of eigenfunctions and quasimodes to geodesic segments in a nonpositively curved manifold in the high frequency limit. This sharpens results of Chen and Sogge as well as Xi and Zhang, which showed that the crux of the problem is to establish bounds on the mixed partials of the distance function on the covering manifold restricted to geodesic segments. The innovation in this work is the development of a formula for the third variation of arc length on the covering manifold, which allows for a coordinate free expressions of these mixed partials.