Smooth metric measure spaces with non-negative curvature (Q429572)

The authors are interested in the geometry and analysis on general smooth metric measure spaces. Here, a \textit{smooth metric measure space} is a triple \((M,g, e^{-f}dv)\) where \((M,g)\) is a Riemannian manifold, \(f\) a smooth function on \(M\) and \(dv\) the volume element induced by the metric \(g\). The term \(e^{-f}dv\) is considered as a weighted measure. The \textit{Bakry-Émery tensor} of the smooth metric measure space \((M,g, e^{-f}dv)\) is defined as NEWLINE\[NEWLINE\mathrm{Ric}_f := \mathrm{Ric} + \mathrm{Hess}(f),NEWLINE\]NEWLINE where \(\mathrm{Ric}\) denotes the Ricci curvature of \(M\) and \(\mathrm{Hess}(f)\) the Hessian of \(f\). The reference for this notion is [\textit{D. Bakry} and \textit{M. Emery}, ``Diffusions hypercontractives'', Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 177--206 (1985; Zbl 0561.60080)]. The authors study some function theoretic and spectral properties of smooth metric measure spaces, and obtain results which have applications to steady gradient Ricci solitons. They assume that \(\mathrm{Ric}_f \geq 0\) and that the growth of \(f\) is linear, that is, NEWLINE\[NEWLINE| f| (x)\leq \alpha r(x) +\betaNEWLINE\]NEWLINE for some constants \(\alpha\) and \(\beta\) where \(r(x)\) is the geodesic distance function from some fixed point. The linear growth \(a\) of \(f\) is then the minumum of such values \(\alpha\).NEWLINENEWLINEThe first result obtained in the paper under review is a gradient estimate for positive \(f\)-harmonic functions on \((M,g, e^{-f}dv)\). As a consequence, the authors prove the strong Liouville property under an optimal sublinear growth assumption on \(f\). They also obtain a sharp upper bound for the bottom spectrum of the \(f\)-Laplacian in terms of the linear growth rate of \(f\). Moreover, they show that if equality holds and \(M\) is not connected at infinity, then \(M\) must be a cylinder. As an application, they obtain that Ricci solitons must be connected at infinity.