Some differential geometric properties of codimension-one foliations of polynomials growth.
It is well known that any codimension-one constant mean curvature foliation of \({\mathbb R}^3\) is totally geodesic. Similar results have been obtained when the ambient manifold is compact, but there are only a few results when the ambient space is noncompact. In the paper under review, the author shows that a codimension-one minimal foliation with growth at most 2 of a complete Riemannian manifold with nonnegative Ricci curvature is totally geodesic (Theorem 1). Here the growth of a foliation \(\mathcal F\) is defined as follows. If \(L\) is a leaf of \(\mathcal F\), the growth of \(L\) is at most \(n\) if there exist positive constants \(a\) and \(b\) so that vol\((B(r))\leq ar^n+b\), for all \(r>0\), where \(B(r)\) is a geodesic ball in \(L\) of radius \(r\). When this is true for all leaves \(L\), the growth of \(\mathcal F\) is said to be at most \(n\). Other main results in the paper are the following. Theorem 2: Let \((M,g)\) be a connected oriented complete Riemannian manifold, and \(\mathcal F\) be a transversely oriented codimension-one foliation on \(M\). If a leaf \(L\in{\mathcal F}\) is of polynomial growth, the following inequality holds \[ \inf_{x\in L}\{\text{Ric}(N,N)(x)+| A| ^2(x)+N(h)(x)\}\leq 0, \] where \(N\) is a unit vector field on \(M\) orthogonal to \(\mathcal F\), Ric is the Ricci curvature of \((M,g)\), \(| A| ^2\) is the square of the norm of the second fundamental form of \(\mathcal F\), and \(h\) is the mean curvature function of \(\mathcal F\). Theorem 3: Let \(\mathcal F\) be a codimension-one minimal foliation of the Euclidean space \({\mathbb R}^{n+1}\). If the growth of \(L\in{\mathcal F}\) is at most \(k\in{\mathbb N}\), then \[ \limsup_{r\to\infty}{\int_{B(r)}| A| ^2\over{r^{k-2}}}<\infty. \]
scientific article; zbMATH DE number 1827637
scientific article