Combinatorial volume preserving flows and taut foliations
The author proves a remarkably strong generalization of a theorem of \textit{R. Roussarie} [Publ. Math., Inst. Hautes Étud. Sci. 43 (1973), 101-141 (1974; Zbl 0356.57017)] and \textit{W. Thurston} [UC Berkeley, Ph. D. Thesis (1972)] to the situation of immersed, incompressible surfaces in closed, tautly foliated 3-manifolds of class \(C^0\). The original theorem concerned imbedded, incompressible surfaces in tautly foliated 3-manifolds of class \(C^2\). The author's result is that, even in the \(C^0\) case, the merely immersed surface is homotopic to an immersed surface that either lies entirely in a leaf or is transverse to the foliation except at isolated saddle tangencies. An elegant proof of the original Roussari-Thurston theorem is due to \textit{J. Hass} [Comment. Math. Helv. 61, 1-32 (1986; Zbl 0601.53024)]. He combines a theorem of \textit{D. Sullivan} [ibid. 54, 218-223 (1979; Zbl 0409.57025)] and a result of \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math. (2) 110, 127-142 (1979; Zbl 0431.53051)]. The first of these theorems guarantees that the tautly foliated manifold admits a Riemannian metric in which each leaf is minimal. The second establishes that a \(\pi_1\)-injective map of a closed, orientable surface into a Riemannian 3-manifold is homotopic to a least area immersion. The author abstracts the ideas of minimal surface theory, Haken's normal surface theory and Thurston's proof of the Sullivan theorem to obtain an elementary and intuitive proof of the \(C^0\) result. An important tool is his notion of a combinatorial, volume-preserving flow.
scientific article; zbMATH DE number 1455390