Minimal surfaces in foliated manifolds
The main object of these studies is a 2-dimensional foliation of a 3- manifold by leaves minimal with respect to some Riemannian metric. Among other results the following considerations are made: Using some local and global properties of minimal surfaces the author collects several topological conditions preventing a foliated compact 3-manifold from the existence of a Riemannian metric making all the leaves minimal. Some of these conditions are well known, some follow immediately from the Novikov theorem. However, the proofs given here are new, interesting and independent of the Novikov theorem which is proved in a new way in the Appendix. Also, the author proves that given a region \(Q\subset R^ 3\) equipped with a minimal foliation \({\mathcal F}\) and a closed curve \(\Gamma\) contained with its convex hull in Q we have \[ \chi (S)\leq k/2+\sum_{x\in A}[1- \deg (x)], \] where S is a branched minimal surface spanned by \(\Gamma\), k is the number of tangencies of \(\Gamma\) to \({\mathcal F}\) and A is the set of branch points of S. The analogous result for higher dimensions is obtained provided that \({\mathcal F}\) is totally geodesic.
scientific article; zbMATH DE number 3969563