Complete foliations of space forms by hypersurfaces
type: Item
instance of: scholarly article
title: Complete foliations of space forms by hypersurfaces
zbMATH Open document ID: 1226.53055
author: D. Kharzeev
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2010-10-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/0908.0786
review text: 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.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 5805927
zbMATH Keywords: graphs
zbMATH Keywords: foliations
zbMATH Keywords: Bernstein-type theorems
zbMATH Keywords: higher order mean curvatures
reviewed by: Paweł Walczak
MaRDI profile type: Publication
OpenAlex ID: W2091539549
arXiv ID: 0908.0786
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DOI: 10.1007/S00574-010-0015-Y
description: scientific article