Immersed surfaces and Dehn surgery

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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 α 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 Λ on T, such that if β is a slope on T with Δ(β, αᵢ) > K for all αᵢ in α, and β is not in Λ, then F remains incompressible after Dehn filling on T along the slope β. 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.