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Limit leaves of an \(H\) lamination are stable. (English) Zbl 1197.53037

Let \(N\) be a Riemannian manifold and \({\mathcal L}\) a codimension one lamination whose leaves have a fixed constant mean curvature (CMC) \(H\in\mathbb{R}\). \({\mathcal L}\) is the union of a collection of pairwise disjoint, connected, injectively immersed hypersurfaces (the leaves of \({\mathcal L}\)) with a certain local product structure. A leaf \(L\) of \({\mathcal L}\) is a limit leaf if \(L\) is contained in the closure of \({\mathcal L}\setminus L\). The authors show that every limit leaf of \({\mathcal L}\) is stable for the Jacobi operator. They prove the following theorem: “The limit leaves of a codimension one \(H\)-lamination of a Riemannian manifold are stable.”
Corollary: “Suppose that \(N\) is a not necessarily complete Riemannian manifold and \({\mathcal L}\) is an \(H\)-lamination of \(N\) with leaves of codimension one. Then, the closure of any collection of its stable leaves has the structure of a sublamination of \({\mathcal L}\), all of whose leaves are stable. Hence, \(\text{Stabl}({\mathcal L})\) has the structure of a minimal lamination of \(N\) and \(\text{Lim}({\mathcal L})\subset\text{Stab}({\mathcal L})\) is a sublamination.” These results have applications to the geometry of embedded minimal and JCMC hypersurfaces in Riemannian manifolds.

MSC:

53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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