Complete foliations of space forms by hypersurfaces
The authors study foliations of space forms by hypersurfaces whose leaves have higher mean curvatures \(S_r\) satisfying some particular conditions. The main results are as follows. Theorem 1. Let \(M^n\subset \mathbb R^{n+1}\) be the graph of a smooth function \(u : \mathbb R^n\to \mathbb R\), such that \(\|\text{grad}(u) - V \|\in L^1(\mathbb R^n)\) for some \(V\in \mathbb R^n\) and \(\|\text{Hess}(u)\|^2 \leq c(1 + \|\text{grad}(u)\|^2)\) for some \(c > 0\). If there exists \(r\) with \(0 \leq r \leq n-1\) such that the functions \(S_{r+1}\) and \(S_{r+2}\) do not change sign on \(M\), then \(M\) has relative nullity \(\nu\geq n - r\). In particular, if \(S_r\neq 0\), then the graph is foliated by hyperplanes of dimension \(n - r\). Theorem 2. There is no smooth, transversely orientable foliation of codimension one of the Euclidean sphere \(S^{n+1}\), whose leaves are complete and have constant scalar curvature greater than one. Theorem 3. Let \(F\) be a smooth, transversely orientable singular foliation of codimension one of \(\mathbb R^{n+1}\), whose leaves are complete, \(r\)-minimal and such that \(S_r\) does not change sign on them. If \(\|X\|\in L^1\) and \(|A|\) is bounded along each leaf, then the relative nullity of each leaf is at least \(n - r\). In particular, if \(S_r\neq 0\) on a leaf, then this leaf is foliated by hyperplanes of dimension \(n - r\). Here, \(X\) is the curvature vector of the 1-dimensional foliation orthogonal to \(F\) and \(A\) is the shape operator of the leaves.
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