A splitting theorem on smooth metric measure spaces
type: Item
instance of: scholarly article
title: A splitting theorem on smooth metric measure spaces
zbMATH Open document ID: 1257.53059
author: Nguyen Thac Dung
published in: Archiv der Mathematik
publication date: 2012-09-18T00:00:00Z
review text: A smooth metric measure space is a Riemannian manifold \(M\) together with a weighted volume form \(e^{-f}\, dv\), where \(f\) is a smooth function on \(M\) and \(dv\) is the volume element associated to the Riemannian metric. The associated weighted Laplacian \(\Delta_f\) is given by \[ \Delta_f u = \Delta u - \langle \nabla f, \nabla u\rangle \,, \] and the associated \(m\)-dimensional Bakry-Émery curvature \(\text{Ric}_m\) is defined by \[ \text{Ric}_m = \text{Ric} - \text{Hess}(f) - \frac{\nabla f \otimes \nabla f}{m-n} \,, \] where \(n\) is the dimension of \(M\) and it is assumed that \(m > n\). To handle the case \(m=n\), the \(n\)-dimensional Bakry-Émery curvature is defined if and only if \(f\) is constant, so in that case \(\text{Ric}_n = \text{Ric} \). Let \(\lambda_1(\Delta_f)\) denote the greatest lower bound for the spectrum of the weighted Laplacian. The main result of the paper states that if (i) \((M,g, e^{-f}\,dv)\) is a complete non-compact smooth metric measure space of dimension \(n\geq 3\), (ii) \(\lambda_1(\Delta_f)>0\), (iii) the \(m\)-dimensional Bakry-Émery curvature satisfies \[ \text{Ric}_m \geq -\frac{m}{m-1} \lambda_1(\Delta_f) \,, \] and (iv) there exists a non-constant \(f\)-harmonic function with finite \(f\)-energy, then \(M\) splits, i.e., \(M={\mathbb R}\times N\) with the warped product metric \( ds^2_M = dt^2 + \eta^2(t)\, ds^2_N \), where \(\eta(t)\) is a positive function and \(N\) is an \((n-1)\)-dimensional manifold. This result generalizes results of \textit{K.-H. Lam} [Trans. Am. Math. Soc. 362, No. 10, 5043--5062 (2010; Zbl 1201.53041)] and of \textit{P. Li} and \textit{J. Wang} [J. Differ. Geom. 58, No. 3, 501--534 (2001; Zbl 1032.58016)].
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 6083810
zbMATH Keywords: Bakry-Émery curvature
zbMATH Keywords: splitting theorem
zbMATH Keywords: weighted Laplacian
zbMATH Keywords: smooth metric measure space
zbMATH Keywords: gradient steady Ricci soliton
reviewed by: Harold Parks
MaRDI profile type: Publication
full work available at URL: s00013-012-0420-0
OpenAlex ID: W1995378021
cites work: Results on a weighted Poincaré inequality of complete manifolds
cites work: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds
cites work: Harmonic functions and the structure of complete manifolds
cites work: Complete manifolds with positive spectrum
cites work: Complete manifolds with positive spectrum. II.
cites work: Smooth metric measure spaces with non-negative curvature
cites work: Analysis of weighted Laplacian and applications to Ricci solitons
DOI: 10.1007/S00013-012-0420-0
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description: scientific article; zbMATH DE number 6083810