Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds. I

TL;DR

This paper proves Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds with limit set Hausdorff dimension less than half the dimension, even with hyperbolic geodesic trapping.

Contribution

It establishes restriction estimates on hyperbolic manifolds with specific geometric conditions, expanding understanding of spectral analysis in such settings.

Findings

Restriction estimate holds when limit set has Hausdorff dimension < n/2

Provides example of restriction estimate with hyperbolic geodesic trapping

Uses spectral measure of the Laplacian for proof

Abstract

In this paper, we investigate the Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds $\Gamma\backslash\mathbb{H}^{n+1}$. Via the spectral measure of the Laplacian, we prove that the Tomas-Stein restriction estimate holds when the limit set has Hausdorff dimension $\delta_\Gamma<n/2$. This provides an example for which restriction estimate holds in the presence of hyperbolic geodesic trapping.