Boundary slopes of immersed surfaces in 3-manifolds. (English) Zbl 0978.57016

Hass, Joel; Rubinstein, J. Hyam; Wang, Shicheng

J. Differ. Geom. 52, No. 2, 303-325 (1999).

A simple loop \(c\) on the boundary of a compact \(3\)-manifold \(M\) is a boundary slope of \(M\), if there is a proper immersion of an essential surface \(F\) into \(M\) such that each component of \(\partial F\) is homotopic to a multiple of \(c\). A fundamental result (proved by Hatcher) is that \(M\) can have only finitely many boundary slopes for embedded essential surfaces. On the other hand, Oertel gave examples of manifolds in which every slope is a boundary slope for an immersed essential surface. This led P. Shalen to ask whether for given \(g\geq 0\) the number of boundary slopes of essential immersed surfaces of genus at most \(g\) is finite. In this paper the authors give a positive answer to Shalen’s question. Moreover in the case that the interior of \(M\) has a complete hyperbolic metric of finite volume they use minimal surface theory to obtain a bound which is a quadratic function of \(g\) and independent of \(M\).

Reviewer: Wolfgang Heil (Tallahassee)

MSC:

57N10

Topology of general \(3\)-manifolds (MSC2010)

Keywords:

boundary slopes