Curves as measured foliation on noncompact surfaces
Let \(M\) be a noncompact orientable surface, and let \(c\) be a simple closed curve which is not homotopic to a disk or to a boundary component. The authors prove that there exists a measured foliation \((F_ c, \mu)\) on \(M\) with the property that \(c\) is a leaf of \(F_ c\), and for every homotopy class \(\alpha\) of simple closed curves on \(M\), we have \(I ((F_ c, \mu), \alpha)= i(c, \alpha)\), where \(I((F_ c, \mu), \alpha)\) is as usual the least upper bound of the set of total mass of curves in the class \(\alpha\) with respect to the measure \(\mu\), and \(i(c,\alpha)\) is the minimum of the number of intersection points with \(c\) of curves in the class \(\alpha\). The result, in the case of compact surfaces, in part of Thurston's theory, and the ingredient which the authors use for the noncompact case is the classification of noncompact surfaces.
scientific article; zbMATH DE number 490571
scientific article