Isolations of geodesic planes in the frame bundle of a hyperbolic $3$-manifold

TL;DR

This paper investigates how geodesic planes in the frame bundle of a hyperbolic 3-manifold are isolated, providing polynomial estimates based on geometric and dynamical quantities.

Contribution

It introduces a quantitative isolation property for lifts of geodesic planes, with estimates depending on geometric and dynamical invariants.

Findings

Isolation estimates are polynomial in tight areas and densities.

Degree of estimates relates to modified critical exponents.

Results apply to geometrically finite hyperbolic 3-manifolds.

Abstract

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.