Immersed surfaces and Dehn surgery

TL;DR

This paper proves that immersed essential surfaces in hyperbolic 3-manifolds largely survive most Dehn fillings, with the proof combining minimal surface theory, differential forms, and geometric group properties.

Contribution

It establishes a finiteness and boundedness result for slopes preserving incompressibility of immersed surfaces after Dehn surgery, extending understanding of surface behavior under manifold modifications.

Findings

Most Dehn fillings preserve the incompressibility of immersed surfaces.

A finite set of slopes can be identified beyond which surfaces remain essential.

Long Freedman tubings of geometrically finite surfaces are also essential.

Abstract

Let $F$ be a proper essential immersed surface in a hyperbolic 3-manifold $M$ with boundary disjoint from a torus boundary component $T$ of $M$. Let $\alpha$ be the set of coannular slopes of $F$ on $T$. The main theorem of the paper shows that there is a constant $K$ and a finite set of slopes $\Lambda$ on $T$, such that if $\beta$ is a slope on $T$ with $\Delta(\beta, \alpha_i) > K$ for all $\alpha_i$ in $\alpha$, and $\beta$ is not in $\Lambda$, then $F$ remains incompressible after Dehn filling on $T$ along the slope $\beta$. In certain sense, this means that $F$ survives most Dehn fillings. The proof uses minimal surface theory, integral of differential forms, and properties of geometrically finite groups. As a consequence of our method, it will also be shown that Freedman tubings of immersed geometrically finite surfaces are essential if the tubes are long enough.