Stability of compact leaves with trivial linear holonomy
The author deals with stability of a compact leaf of a foliation under \(C^ 1\)-perturbation. In the first place, he proves the following Theorem 1, which is a generalization of the results of \textit{W. Thurston} [Topology 13, 347-352 (1974; Zbl 0305.57025)] and of \textit{R. Langevin} and \textit{H. Rosenberg} [Topology 16, 107-111 (1977; Zbl 0346.57009)], which are also generalizations of the Reeb stability theorem. Let \(F_ 0\) denote a \(C^ 1\)-codimension q foliation of a compact smooth manifold M and \(L_ 0\) a compact leaf of \(F_ 0\). Theorem 1. Let \(g_ 1,...,g_ k\) be loops in \(L_ 0\) based at \(x_ 0\), whose real homology classes generate \(H_ 1(L_ 0;{\mathbb{R}})\). Suppose \(L_ 0\) has trivial linear holonomy. Then for any neighbourhood N of \(L_ 0\subset M\) there exist neighbourhoods V of \(F_ 0\) in \(Fol_ qM\) and U of \(x_ 0\) in the transverse disk \(D(x_ 0)\) such that for all F in V, (1) the perturbed holonomy \(H_ F(q_ i): U\to D(x_ 0)\) is defined \((i=1,...,k)\); (2) if for some \(x\in U\), \(H_ F(g_ i)(x)=x\) (for all i) then the leaf L of F passing through x is contained in N, diffeomorphic to \(L_ 0\) and hence compact. As applications of the above theorem, the author obtains several sufficient conditions under which compact leaves are stable. Let p: \(M\to B\) be a smooth fibration of M by the fibers of p. The fibers of p are said stable in the large if every foliation F of M sufficiently \(C^ 1\)- close to \(F_ 0\) \((=foliation\) by the fibers of p) has a compact leaf near to the one of the original fibers. A celebrated theorem of Seifert states that the fibers of the Hopf fibration \(S^ 3\to S^ 2\) are stable in the large. With the exception of {\S}7, most of the theorems in this paper concern this notion of stability in the large. We quote one of them only because it is easily understood. Theorem. Let p: \(M\to S^ 1\) be a fibration obtained as a suspension of a diffeomorphism f: \(L\to L\). If \(H_ 1(L;{\mathbb{R}})\cong {\mathbb{R}}\) and \(f_*: H_ 1(L;{\mathbb{R}})\to H_ 1(L;{\mathbb{R}})\) is a multiplication by a negative number, then L is stable in the large. In the appendix, the author proves several fundamental facts concerning stability arguments including the details of some facts which other authors used implicitly before. In this sense, the paper is fairly self- contained and will be a good reference in this direction of research.
stability of a compact leaf of a foliation
scientific article; zbMATH DE number 4119262