Some differential geometric properties of codimension-one foliations of polynomials growth.
type: Item
instance of: scholarly article
title: Some differential geometric properties of codimension-one foliations of polynomials growth.
zbMATH Open document ID: 1034.53027
author: Gen-ichi Oshikiri
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2002-01-01T00:00:00Z
review text: 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 \[ \limsup_{r\to\infty}{\int_{B(r)}| A| ^2\over{r^{k-2}}}<\infty. \]
reviewed by: Pascual Lucas Saorín
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 1827637
zbMATH Keywords: minimal foliation
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: polynomial growth
zbMATH Keywords: Ricci curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113247570
OpenAlex ID: W1985305801
Wikidata QID: Q115227298
cites work: Hypersurfaces of constant mean curvature with finite index and volume of polynomial growth
cites work: Erratum to ``Hypersurfaces of constant mean curvature with finite index and volume of polynomial growth''
cites work: Stability of hypersurfaces with constant mean curvature
cites work: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds
cites work: Foliations by hypersurfaces with constant mean curvature
cites work: Minimal cones and the Bernstein problem
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Q3041926
cites work: Elliptic Partial Differential Equations of Second Order
cites work: The decomposition of a differentiable manifold and its applications
cites work: Q3873112
cites work: The topology and geometry of embedded surfaces of constant mean curvature
cites work: Una maggiovazione integrale per le curvature delle ipersuperfici minimali
cites work: A remark on minimal foliations
cites work: Jacobi fields and the stability of leaves of codimension-one minimal foliations
cites work: Totally geodesic foliations and Killing fields
cites work: Structural equations and an integral formula for foliated manifolds
cites work: Q4743396
cites work: Q3732470
cites work: On foliations of \({\mathbb{R}}^{n+1}\) by minimal hypersurfaces
cites work: Q4337693
DOI: 10.2748/TMJ/1113247570
description: scientific article; zbMATH DE number 1827637