A splitting theorem on smooth metric measure spaces
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)].
scientific article; zbMATH DE number 6083810
scientific article