Immersed surfaces and Dehn surgery

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.